Paper Title: Entropy, Thermodynamics and the Geometrization of the Language Model.



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    Entropy, Thermodynamics and the Geometrization of the Language Model
        
    

    Wenzhe Yang
    
    

    
    
    

    
    <https://orcid.org/0000-0002-2456-7453>

    
    



§ ABSTRACT 

In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. 

Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold.








Keywords: Language Model, Moduli Space, Entropy Function, Thermodynamics, Geometrization of Language Model, Boltzmann Manifold.

 




§ INTRODUCTION


In November 2022, OpenAI surprised the world with the release of the highly “intelligent” ChatGPT, which possesses many striking abilities that far exceed any other available systems <cit.>. For example, it can answer questions with a very high accuracy and engage in human-like conversations. It is the first time that a computer performs well enough in human languages that makes people wonder whether it acquires true intelligence. Ever since then, the arm race in the area of Large Language Model (LLM) has become extremely fierce, with the announcements of new LLMs with hundreds of billions of parameters regularly. Now there are abundant Chatbots based on LLMs available in markets, many of which have astonishing new abilities and behaviors <cit.>. Especially, when the number of parameters (e.g., weights) are increased to a threshold, then some surprising behaviors emerges <cit.>, which is extremely intriguing. 

An LLM is a generative model, which means it can practically sample a new sentence given a prompt. More precisely, suppose we are given an arbitrary prompt w_1⋯ w_n, where each w_i is a word, an LLM can compute the probability distribution of the next word, i.e., P(w_n+1|w_1 ⋯ w_n). Next, we sample the next word w_n+1 according to this probability distribution <cit.>. After we obtain w_n+1, we repeat this process and sample w_n+2 according to the probability distribution P(w_n+2|w_1 ⋯ w_n w_n+1) computed by the LLM. We repeat this process until some conditions are met, e.g., the LLM outputs an end-of-sequence symbol.

Let us now briefly review the basic structures of the LLMs based on the transformer architecture, to the extent that will be needed in this paper. For a more detailed and thorough treatment, the readers are referred to the book <cit.>. First, in order for a computer to be able to compute words, words are first embedded into a vector space, which is called the Word-Embedding <cit.>. Suppose 𝒲 is the finite set of all words, a Word-Embedding is an injective map

    ι: 𝒲→ℝ^N,

where the standard inner product ⟨·, ·⟩ on ℝ^N measures the correlations between words. Given an arbitrary vector 𝐯∈ℝ^N, it defines a distribution on 𝒲 via

    P_𝐯(w) = exp( ⟨𝐯, ι(w) ⟩)/∑_w' ∈𝒲exp( ⟨𝐯, ι(w') ⟩),

which is called the Boltzmann distribution in statistical mechanics <cit.>. 


In particular, given a word w ∈𝒲, it also gives us a distribution on 𝒲 via

    P_ι(w)(w') = exp( ⟨ι(w), ι(w') ⟩)/∑_w”∈𝒲exp( ⟨ι(w), ι(w”) ⟩),

the value of which measures the correlations between w and w'. In practice, the embedding ι is chosen such that P_ι(w)(w') is the probability that w' is in an l-neighborhood of w <cit.>. One nice property of this embedding is that generally the embedded vectors {ι(w)| w ∈𝒲} lie in a lower dimensional vector subspace of ℝ^N as these vectors are not independent from each other. A famous linear relation is 

    ι(Queen) - ι(Woman) = ι(King) - ι(Man).

For more details, the readers are referred to the papers <cit.>.

After the Word-Embedding, a sentence w_1⋯ w_n is mapped to a vector

    (ι(w_1),⋯,ι(w_n)) ∈ (ℝ^N)^n.

As the name suggests, a transformer is a transformation

    T_n: (ℝ^N)^n → (ℝ^N)^n,

which is a differentiable often smooth map <cit.>. For the details of the construction of a transformer and some of the ideas behind it, the readers are referred to the paper <cit.> and the book <cit.>. In particular, each transformer depends on internal parameters which are tuned during the training process. Notice that transformers can be nested in the form

    𝒯_n = T_n,k∘⋯∘ T_n,1,

where k is the number of layers and different layers have different parameters. Let Proj be the projection of (ℝ^N)^n to the last ℝ^N in the direct product, hence Proj∘𝒯(ι(w_1),⋯,ι(w_n)) is a vector of ℝ^N, which defines a distribution on 𝒲 via Eq. (<ref>). The training of the transformers on a very large corpse is to make sure that this distribution is a good approximation of P(·|w_1⋯ w_n) <cit.>.

The LLMs have made rapid advance from a numerical and experimental point of view, but there have been very few papers that attempt to understand the properties and behaviors of LLMs from a analytical point of view. More precisely, we now desperately need a theory that can explain these surprising properties and behaviors, which is the main motivation of this paper. 


In this paper, we first use set theory, analysis and probability theory to rigorously define the concept of Causal Language Model (CLM), and show it is equivalent to the Predicative Language Model (PLM), which basically predicts the next word of a prompt. We formulate a generalized distributional hypothesis for a CLM based on functional analysis and topology. We introduce the concept of the moduli space of distributions on words, which is the key metric space to understand the power and properties of a language model.

The development of human language is for human to exchange information efficiently between individuals. In order to analytically study a language model, we must understand how information is quantified during conversations. In this paper, we introduce the entropy function for CLM and PLM, and show how it can play a crucial role in the study of language models. In particular, we analyze the zero points of the entropy function, and show how to use information theory to understand the meaning of them. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. We also formulate an extension conjecture about the properties of AGI using the entropy function.

Next, we use thermodynamics to give an physical interpretation of the language models. In this interpretation, a sentence is a microstate that has a potential energy. All the possible outputs of a prompt define a statistical ensemble, which is distributed according to the Boltzmann distribution <cit.>. We introduce the concept of the partition function, internal energy and Helmholtz free energy for this statistical ensemble associated. Using the Helmholtz free energy, we show how the game of word-predicting can be interpreted as the process of a molecule growing in physics.

Later, we formulate the concept of the geometrization of a language model, and show that the accuracy of a language model is determined by how precise the embedding of the moduli space in a geometrization is. We also define the Word-Embedding with respect to a general manifold with a pairing structure, and introduce the corresponding Boltzmann manifold with respect to it. At last, we show how the current LLMs can be viewed as special cases of the geometrization of language models that use the linear space together with its inner product. Based on our theory, we also pose several important open questions, e.g., which manifold together with a pairing is optimal for the geometrization of language models. Perhaps different languages would need different manifolds and pairings, namely different Boltzmann manifolds in the geometrization. Another important direction is whether the current researches on statistical mechanics and differential geometry can offer new tools in the study of LLMs.


The layout of this paper is as follows. In Section <ref>, we use set theory, analysis and probability theory to rigorously define language models, i.e., CLM and PLM, and we introduce the moduli space of distributions associated with a language model. In Section <ref>, we introduce the entropy function, and show how information theory can be applied to the study of language models. In Section <ref>, we present a thermodynamic interpretation of language models and define statistical functions such as partition function, internal energy and Helmholtz free energy for a language model. In Section <ref>, we formulate the concept of the geometrization of language models, and define the Boltzmann manifold. In Section <ref>, we show that the current LLMs are special cases of the geometrization of language models. In Section <ref>, we conclude this paper and propose several open problems.









§ THE LANGUAGE MODEL AND MODULI SPACE
 



In this section, we will use set theory, analysis and probability theory to give rigorous definitions of language models. We first define what is a general sentence in the sense of set theory <cit.>. Then we introduce the definitions of the General Language Model (GLM), the Causal Language Model (CLM) and the Predicative Language Model (PLM). At last, we define what is called the moduli space of distributions for a PLM, which is the crucial metric space to understand the properties and behaviors of language models.



 §.§ Sentence as a Free Mathematical Sequence?


Suppose 𝒲 is the finite set of all words in a language, the set of all sentences associated with 𝒲 is defined in the same way as in set theory and logic <cit.>. First, let ∅ be the unique sentence with no word, whose length is by definition 0. Let 𝒮_0 be the set that consists of a unique element ∅

    𝒮_0 = {∅}.


The inclusion of ∅ as a sentence will make our analysis simpler from a mathematical point of view. While later we will also see ∅ is a suitable symbol for the end (or even the beginning) of a sentence.

A sentence of length 1 is just a word of 𝒲. Let 𝒮_1 be the set of all sentences of length 1, namely

    𝒮_1 = 𝒲.

Given an positive integer n ≥ 2, let 𝒮_n be the set of all sentences of length n, i.e., an element of 𝒮_n is a sequence

    𝐬_n=w_1 w_2 ⋯ w_n with w_i ∈𝒲.

Here, the words in a sentence do not need to be different from each other, e.g., we allow the case where w_i=w_j, i ≠ j. The set of all sentences of finite lengths is by definition the union of all 𝒮_n

    𝒮 = ⋃_n=0^∞𝒮_n.

Notice that here we allow the length of a sentence to be arbitrarily large. 

In the definition of 𝒮, we have allowed the existences of arbitrary sentences that do not satisfy grammar rules. In this paper, the grammar is instead considered as part of the language model.



The cardinality of 𝒮 is countably infinite, i.e., there exists a bijective map between 𝒮 and ℕ.


Since 𝒲 is a finite set, this lemma is an immediate result of elementary set theory, see the book <cit.> for more details.



Even in this most general setting, we do not consider the case of sentence of infinite length, i.e., infinite sequence, which is very important from both practical and set-theoretic points of view. Practically, human brain has not evolved to understand infinity properly, so it makes no sense to a mortal that a sentence is infinite. Set-theoretically, the set of all infinite sequences has the same cardinality as the real numbers, i.e., its cardinality is ℵ_1 <cit.>. So we do not really want to walk into the deep water of axiomatic set theory.


There is a natural operation defined on 𝒮 that is called concatenation: given a sentence s_m ∈𝒮_m and a sentence s_n ∈𝒮_n, their concatenation is a new sentence s_m s_n that lies in 𝒮_m+n. As a convention, the concatenation between ∅ and s_n is s_n itself, i.e.,

    ∅𝐬_n = 𝐬_n ∅ = 𝐬_n.

This concatenation operation is associative, but not commutative, while the empty sentence ∅ serves as a two-sided identity. So the set 𝒮 has a monoid structure <cit.>.








 §.§ The Causal Language Model


Now we are ready to give a more mathematical and general definition of the language model.


 
Given an arbitrary sentence s ∈𝒮, a General Language Model (GLM) ℒ_G defines a joint probability distribution P_ℒ_G,𝐬 on 𝒮×𝒮, where the probability P_ℒ_G,𝐬(𝐬_1,𝐬_2) measures the likelihood of the concatenation 𝐬_1𝐬𝐬_2 among all possible choices. Namely, P_ℒ_G,𝐬(𝐬_1,𝐬_2) is the probability that there is a sentence 𝐬_1 before 𝐬 and a sentence 𝐬_2 after 𝐬.



Now, we consider the case where the sentence 𝐬_1 on the left side of 𝐬 is ∅, in which case the probability distribution in Definition <ref> defines a distribution on 𝒮. Hence it motivates the following definition.


Given an arbitrary sentence 𝐬∈𝒮, a Causal Language Model (CLM) ℒ defines a probability distribution on 𝒮. Namely, given a prompt 𝐬, we have a distribution P(·|𝐬) such that the probability that ℒ outputs a sentence 𝐬' is P_ℒ(𝐬'|𝐬).


In the definition, we allow the prompt 𝐬 to be ∅, in which case the CLM ℒ defines a probability distribution on 𝒮, i.e., P(𝐬_out|∅). It is called causal because 𝐬' is after 𝐬, so there is a causality between 𝐬 and 𝐬'. From the definition, the input to ℒ can be of arbitrary length, while the output sentence can also be of arbitrary length. In fact, such a definition of CLM is extremely powerful.




Let ℬ(𝒮,ℝ) be the set of all functions with finite absolute summation, i.e.,

    ℬ(𝒮,ℝ) = {f:𝒮→ℝ| ∑_𝐬∈𝒮|f(𝐬)| < ∞}.

The set ℬ(𝒮,ℝ) is a metric space with a metric given by

    ‖ f-g ‖  =  ∑_𝐬∈𝒮 | f(𝐬) - g(𝐬) |.

For more details about this space, the readers are referred to the book <cit.>. Clearly for every 𝐬, the distribution P_ℒ(·|𝐬) is a non-negative function that lies in ℬ(𝒮,ℝ), in fact ‖ f ‖ = 1. Therefore, there exists a map

    ρ_ℒ: 𝒮→ℬ(𝒮,ℝ).


In 1950s, linguists formulated the distributional hypothesis, which basically says that the meaning of words are determined by their contexts <cit.>. We generalize this hypothesis using real analysis.

If ℒ is an intelligent language model, then the map ρ_ℒ is an embedding that is also discrete. Namely, each point ρ_ℒ(𝐬) has a neighborhood that does not contain any other ρ_ℒ(𝐬') where 𝐬≠𝐬'. 

For more details of the meaning of “discrete” in the sense of topology, the readers are referred to the book <cit.>. As a result, the embedding ρ_ℒ induces a metric on 𝒮.



It should be stressed that in the definition of a CLM, we do not require it to have “intelligence” behaviors. For example, for a given prompt 𝐬, the probability distribution P(· | 𝐬) can be a “random” distribution over 𝒮, which can even outputs sentences that are quite pathological. Hence, intelligence is not incorporated in the definition of the language model. Instead, only highly special language models can exhibit intelligence behaviors.


 







 §.§ The Predicative Language Model
 

Now we give the definition of the Predicative Language Model.


Given a sentence 𝐬, a Predicative Language Model (PLM) ℒ_P defines a probability distribution P(·|𝐬) on 𝒲∪∅. Here, P(∅|𝐬) is the probability that ℒ_P outputs nothing and P(w|𝐬) is the probability that ℒ_P outputs the word w ∈𝒲.


We have the following lemma.

The definition of the PLM is equivalent to the definition of the CLM.


If we have a PLM ℒ_P, then by repetitively using it we can generate sentences. More precisely, given an input 𝐬∈𝒮, we can generate a sentence 𝐬'=w_1w_2⋯ w_n until ℒ_P outputs ∅ via

    ℒ_P(𝐬) = w_1,  ⋯,  ℒ_P(𝐬w_1 ⋯ w_n-1) = w_n,  ℒ_P(𝐬w_1 ⋯ w_n) = ∅.

The probability that ℒ_P outputs 𝐬'=w_1⋯ w_n given 𝐬 is 

    P(𝐬'|𝐬)=P(w_1|𝐬) · P(w_2|𝐬w_1) ·…· P(w_n|𝐬w_1 ⋯ w_n-1) ·  P(∅|𝐬𝐬').

In this way, the PLM ℒ_P defines a probability distribution on 𝒮 for any prompt 𝐬, hence it produces a CLM.

We now show the inverse construction from CLM to PLM. Namely, suppose we have a CLM ℒ, we now construct a PLM ℒ_P that reverses the process in the previous paragraph. Given a prompt 𝐬, let us construct a distribution on 𝒲∪∅. First, P(∅|𝐬) is straightforward to define, i.e., P_ℒ(∅|𝐬). For an arbitrary word w, let 
    P(w|𝐬) = ∑_𝐬' ∈𝒮 P_ℒ(w𝐬'|𝐬).

Notice that here the sum is just over all sentences whose first word is w. In this way, we obtain a PLM.



As shown in the proof of this lemma, the empty sentence ∅, which is the unit of the monoid 𝒮, plays the role of ending the sentence outputted by ℒ_P. Hence comes a crucial definition in our treatment of PLM.

 
The space of all possible distributions on 𝒲∪{∅} that come from a PLM ℒ_P is called the moduli space of distributions of ℒ_P, which is expressed as

    ℳ = { P(·| 𝐬)| 𝐬∈𝒮}.


In fact, this moduli space is a metric space. Let ℬ(𝒲∪{∅},ℝ) be the set of all functions with finite absolute summation

    ℬ(𝒲∪{∅},ℝ) = {f:𝒲∪{∅}→ℝ| ∑_w ∈𝒲∪{∅}|f(w)| < ∞}.

The set ℬ(𝒲∪{∅},ℝ) is a metric space with a metric given by

    ‖ f-g ‖  =  ∑_w∈𝒲∪{∅} | f(w) - g(w) |.

For more details, the readers are referred to the book <cit.>. Clearly, for every 𝐬∈𝒮, the distribution P(·|𝐬) ∈ℳ is a non-negative function that lies in ℬ(𝒲∪{∅},ℝ), in fact ‖ P(·|𝐬) ‖ = 1. Therefore, ‖·‖ induces a metric d on the moduli space ℳ

    d: ℳ×ℳ→ℝ.


The moduli space ℳ together with the metric d is the key object to understand a PLM.








§ THE ENTROPY FUNCTION
 


In this section, we introduce the concept of the entropy function and show how it quantifies the information contained in the output for a given prompt. We will study the zero-points, which are also the minimal points of the entropy function, and discuss their crucial properties. We argue that these zero points and the points where the entropy is close to 0 are the key obstacles for a LLM to approximate an intelligent language model. At last, we formulate a conjecture regarding AGI.




 §.§ Information Flow and Entropy


The entropy function for a CLM ℒ is defined just as in statistical mechanics and information theory <cit.>.


Given a prompt s ∈𝒮, the entropy of 𝐬 with respect to a CLM ℒ is given by

    S_ℒ(𝐬)= - ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) log P_ℒ(𝐬'|𝐬),

where the sum is over all possible outputs 𝐬'.

We will see that the entropy function is a key mathematical concept to analyze the behaviors of language models. From the definition, the entropy function is a non-negative function defined on 𝒮

    S_ℒ: 𝒮→ℝ.

As a special example, if the prompt is ∅, i.e., no prompt, the entropy associated with it has a special name which is stated as a definition.

The background entropy of a CLM ℒ is

    S_ℒ(∅) = - ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|∅) log P_ℒ(𝐬'|∅).


The background entropy S_ℒ(∅) is a quantity that gives a zeroth order measurement about how complicated a language is. If S_ℒ(∅) is very large, this intuitively mean that the CLM is quite diverse. While if S_ℒ(∅) is very small, it intuitively means that the CLM is very rigid.


Since the development of language is for individuals to exchange information, so a language model itself must be able to demonstrate this. Here we argue that the definition of the entropy function exactly quantify the information during this process <cit.>. More precisely, given a prompt 𝐬, the entropy S_ℒ(𝐬) measures how vague are the answers to the prompt 𝐬 with respect to this CLM ℒ. If S(𝐬) is very large, this means that the answers can be very diverse. On the other hand, if S(𝐬) is very small, this means the answer to 𝐬 is very rigid. Therefore, after the CLM outputs 𝐬', we can say the information revealed by 𝐬' is S_ℒ(𝐬) <cit.>.


Just like statistical mechanics, we argue that the most important function for a CLM is the entropy function <cit.>.








 §.§ The Zero Points of the Entropy Function






Let us now look at the zero points of S_ℒ. If S_ℒ(𝐬)=0, then it means that the answer to the prompt 𝐬 is unique in the CLM ℒ, i.e., there exists a unique sentence 𝐬' ∈𝒮 such that P_ℒ(𝐬'|𝐬) = 1. Hence there is the interpretation that the CLM ℒ detects that the answer to the prompt 𝐬 is unique, and thus the CLM itself actually contain this information. For example, suppose the prompt 𝐬 is

    𝐬 =

If the entropy function of an intelligent CLM ℒ vanishes at 𝐬, i.e., S_ℒ(𝐬) = 0, then this means there is only one possible output of ℒ for the prompt 𝐬. Of course, an intelligent ℒ that knows about human history will output 

    𝐬' =

with probability 1. It should be noticed that at the point 𝐬, the vanishing of the entropy only implies there is only one possible output to 𝐬. Another CLM ℒ_1 could still have S_ℒ_1(𝐬)=0, but it instead outputs 

    𝐬” = .

We say the CLM ℒ_1 outputs a `wrong' answer because Gauss is a mathematician, so the information given by ℒ_1 is wrong. But `wrong' information is still information, the entropy function itself only quantifies the amount of information, but it cannot detect whether the information contained is right or wrong, which needs to be checked with what happens in the outside world. 

Given a CLM ℒ, a zero point 𝐬 of the entropy function S_ℒ is called a singularity of ℒ.

Hence the conclusion is that a CLM memorize information at its singularities. We argue that the existence of these singularities and the points where the entropy function is very small are the key obstacle for an LLM to approximate an intelligent CLM. 






 §.§ An AGI Conjecture
 


It is hard to define which CLM is Artificial General Intelligence (AGI) mathematically! Our idea is that if a CLM ℒ is an AGI, then it must satisfy certain mathematical properties. In this paper, we formulate a conjecture about this.








 
If a CLM is an AGI, then there exists a finite subset 𝒮_AGI⊂𝒮 such that the entropy function S_ℒ: 𝒮→ℝ is determined by its value on 𝒮_AGI.

The motivation of this conjecture comes from modern mathematics. In modern mathematics, we first give a finite set of definitions and axioms, and then mathematics is about proof, which uses these finite set of rules of prove a result that is not known a priori. The key motivation for Conjecture <ref> is that the value of S_ℒ on 𝒮 is determined using mathematical deductions by its value on the finite set 𝒮_AGI <cit.>. 




§ A THERMODYNAMIC INTERPRETATION OF LANGUAGE MODELS
 

In this section, we will give a thermodynamic interpretation of the CLM defined in Section <ref>. We show how the concepts and properties of the CLM naturally admit physical interpretations. Then we will define thermodynamic functions such as internal energy, Helmholtz free energy on CLM, and show how they help us understand phenomena in language models. At last, we give a physical interpretation to how a sentence “grows” in a PLM.




 §.§ The Boltzmann Distribution


First, let us introduce the Boltzman distribution in statistical mechanics <cit.>. Suppose we are given a physical system that can occupy different microstates, and suppose the energy of the microstate i is E_i. Let k be the Boltzmann constant and T be the temperature, then the probability that the system occupies  microstate i satisfies

    p_i ∝exp( - E_i/kT).

Let the partition function of the system be

    𝒵= ∑_iexp( - E_i/kT),

where the sum is over all possible microstates. Then p_i is just

    p_i = 1/𝒵·exp( - E_i/kT),

which is called the Boltzman distribution. The Boltzmann distribution is the distribution that maximizes the entropy

    S = - ∑_i p_i log p_i

subject to the constraint ∑ p_i =1 and ∑_i p_i E_i is a constant, i.e., the average energy is kept fixed.



 §.§ A Thermodynamic Interpretation
 

We now give a thermodynamic interpretation of CLM. 

In a CLM ℒ, a sentence is a microstate and for every sentence 𝐬∈𝒮, there is an energy E_ℒ(𝐬) associated with it.

The energy of ∅ is called the vacuum energy, and the energy of a word w ∈𝒲 is called its self-energy <cit.>.


As in physics, a word w ∈𝒲 itself can have self-interaction, it has an internal energy E(w), the meaning of this internal energy is that given a random sentence 𝐬, the probability that the first word of 𝐬 is w is proportional to exp(-β E(w) ). Here β is the inverse of temperature.


Given a length-2 sentence w_1w_2, its energy E_ℒ(w_1w_2) is the interaction between the two words w_1 and w_2, and E_ℒ(w_1w_2) is lower if the interaction between w_1 and w_2 is stronger. 

Since a language is essentially a discrete object, a word does not have a continuous degree of freedom. Therefore, there is no kinetic energy for a word, and only potential energy exists when words form a sentence. 

Here comes the definition of the statistical ensemble in a language model.

For a given prompt 𝐬, the statistical ensemble associated with it is 

    {𝐬𝐬'|𝐬' ∈𝒮}.



For later purpose, we introduce another quantity β which is the inverse of temperature <cit.>. For a fixed CLM, the value of β is usually fixed to be 1. 

In a CLM ℒ, for a given prompt 𝐬 the probability that ℒ outputs 𝐬' satisfies

    P_ℒ(𝐬'|𝐬) ∝exp(-β E_ℒ(𝐬𝐬') ),

where E_ℒ(𝐬𝐬') is the energy of the concatenation 𝐬𝐬'.   


More precisely, for a given prompt 𝐬, the probability that ℒ outputs 𝐬' is

    P_ℒ(𝐬'|𝐬) = 1/𝒵_ℒ·exp( - β E_ℒ(𝐬𝐬') ),

where E_ℒ(𝐬𝐬') is the energy of the concatenation 𝐬𝐬' and 𝒵_ℒ is the partition function associated with 𝐬

    𝒵_ℒ(𝐬) = ∑_𝐬' ∈𝒮exp( - β E_ℒ(𝐬𝐬')).

The partition function is in fact a thermodynamic function defined on 𝒮

    𝒵_ℒ:𝒮→ℝ.

 

The average energy of the statistical ensemble associated with 𝐬 is 

    U_ℒ (𝐬) = ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) E_ℒ(𝐬𝐬'),

which is called the internal energy of the ensemble. The internal energy U_ℒ is also a well-defined function on 𝒮

    U_ℒ : 𝒮→ℝ.

But notice that U_ℒ is different from E_ℒ: E_ℒ(𝐬) is the energy of the microstate 𝐬, while U_ℒ(𝐬) is the average energy of the statistical ensemble associated with 𝐬. Recall that the entropy for the statistical ensemble associated to 𝐬 is

    S_ℒ(𝐬)=-  ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) log  P_ℒ(𝐬'|𝐬) ,

which is simplified to

    S_ℒ = β U_ℒ + log𝒵_ℒ.

The Helmholtz free energy is given by

    F_ℒ = U_ℒ - 1/β S_ℒ=- 1/βlog𝒵_ℒ.

The Helmholtz free energy determines whether a sentence 𝐬 is likely to be followed by another sentence or not.







 §.§ A Physical Picture of How a Sentence Grows



Let us now interpret the process of predicting the next word in a PLM using how molecule grows in statistical mechanics. First, because of the equivalence between CLM and PLM, the interpretation in Section <ref> admits immediate generalization to PLM. In this physical picture, a word is view as an atom, and a sentence is a sequence of atoms, whose energy depends on the order of these atoms. Suppose we are given a prompt 𝐬∈𝒮_n, which is composed of n atoms. To predict the next word is to add an atom to 𝐬. For a word w ∈𝒲∪{∅}, the energy of 𝐬w is E(𝐬w), and the internal energy is by definition

    ∑_w ∈𝒲∪{∅} P(w|𝐬) E(𝐬w).

We also have the entropy

    - ∑_w ∈𝒲∪{∅} P(w|𝐬) log P(w|𝐬).

There are two trends that fight each other

    
  * the internal energy (Eq. (<ref>)) wants to be as small as possible,
    
  * the entropy (Eq. (<ref>)) wants to be as large as possible.

As a result, the Helmholtz free energy (Eq. (<ref>)) is minimized, and the resulting distribution is the Boltzmann distribution. After sampling the next word, say w_n+1, we can continue this process with the new sentence 𝐬w_n+1 and sample its next one. This process is highly similar to the process of how a molecule grow in statistical mechanics!


But there remains the question of how to compute the energy of a sentence. A sentence can be very long, hence the computation of its energy is a typical many-body problem. The power of LLM is to approximately compute the interaction energy between the “atoms” in a sentence using machine learning.




§ THE GEOMETRIZATION OF LANGUAGE MODELS
 

In this section, we will introduce the concept of the geometrization of a PLM, which lays the foundation for studying language models using differential geometry. We will also define a more general version of the Word-Embedding, which leads to the concept of a Boltzmann manifold.





 §.§ The Geometrization of PLM


Suppose we have a PLM ℒ_P, whose moduli space of distributions is ℳ (Definition <ref>). Moreover, we have a map

    ρ: 𝒮→ℳ,

where ρ sends a sentence 𝐬∈𝒮 to the distribution P(·| 𝐬) ∈ℳ. Notice that ℳ is a metric space, the details of which can be found in Section <ref>.

Generally, the moduli space ℳ itself is extremely complicated, to study which we need to borrow ideas from both mathematics and physics. The first step is to geometrize this complicated metric space, which “prompts” the following definition.


The ϵ-geometrization of a PLM is a manifold M and a continuous map

    Ψ: ℳ→ M,

where each point of M represents a probability distribution on 𝒲∪∅ that is also computable. Moreover, for every point P(·| 𝐬) of ℳ, the norm of the difference ‖Ψ(P(·| 𝐬))-P(·| 𝐬) ‖ < ϵ(n). Here, the error ϵ(n) can depend on the length of the sentence 𝐬∈𝒮_n.

Notice that the norm ‖·‖ is defined in the metric space ℬ(𝒲∪{∅},ℝ) (Eq. (<ref>)). The composition Ψ∘ρ is a map from 𝒮 to M

    Ψ∘ρ: 𝒮→ M.

The motivation of the geometrization of the moduli space ℳ is that we try to find a finite dimensional manifold M to approximately describe the infinite set ℳ. Then we use the geometry of this manifold to study the bizarre metric space ℳ.




 §.§ The Word-Embedding and the Boltzmann Manifold
 

One way to construct an ϵ-geometrization is via an additional structure on a manifold called the pairing. Suppose M is a manifold, a symmetric pairing is a smooth function 

    ⟨·, ·⟩ : M × M →ℝ,

which maps any two points pt_1, pt_2 ∈ M to a number in ℝ. Here “symmetric” means

    ⟨pt_1, pt_2 ⟩ = ⟨pt_2, pt_1 ⟩.

A Word-Embedding is an injective map

    ι: 𝒲∪{∅}→ M.

Given an arbitrary point pt∈ M, it defines a distribution P_pt on 𝒲∪{∅} via

    P_pt(w) =  exp( - β⟨ pt,ι(w) ⟩)/∑_w' ∈𝒲∪∅exp( - β⟨ pt,ι(w') ⟩),

which is exactly from the Boltzmann distribution <cit.>. Hence every point of M defines a distribution on 𝒲∪{∅} and M is a space of distributions on 𝒲∪{∅}.


The manifold M together with the pairing ⟨·,·⟩ and the Word-Embedding ι is called a Boltzmann manifold. 


Since every point of the Boltzmann manifold M defines a distribution on 𝒲∪{∅}, M is a subspace of  ℬ(𝒲∪{∅},ℝ) (Eq. (<ref>)). Therefore, given two points pt and pt' of M, their distance is

    d(pt,pt') =   ‖ P_pt-P_pt'‖.

This distance function induces a metric g_B on M, which is called the Boltzmann metric, hence M is in fact a Riemannian manifold <cit.>.

Now suppose M gives us an ϵ-geometrization with the Moduli-Embedding Ψ: ℳ→ M. Since 𝒮_n is the direct product 𝒲^n, thus if we believe the map Ψ∘ρ has sufficiently nice properties, then there exists a diagram

    𝒮_n = 𝒲^n @>ι^n>> M^n
    
    @VVρV @VVΛ_nV
    ℳ @>Ψ>> M
      ,

which is commutative up to a small error that depends on n. More precisely, we assume there exists a smooth map Λ_n:M^n → M such that for every 𝐬=w_1w_2⋯ w_n, we have

    Ψ∘ρ (𝐬) = Λ_n( ι(w_1), ⋯, ι(w_n) ).

It means that the map ρ∘Ψ is determined by the map ι and {Λ_n:n ≥ 2 }. The point is that the map Ψ∘ρ is defined on the infinite set 𝒮, which can be highly complicated. While the map ι is defined on a finite set and Λ_n is defined on a smooth manifold, hence can be studied by the modern theory of differential geometry <cit.>.









§ LLM AS A SPECIAL GEOMETRIZATION
 

In this section, we show that the current LLMs based on the transformer architecture are special cases of geometrization in the sense of Section <ref>. We also introduce the fundamental thermodynamic relation of LLMs.




 §.§ The Geometrization and LLM



The Boltzmann manifold used in LLMs is the simplest one: ℝ^N together with the standard inner product ⟨· , ·⟩. In practice, first we obtain a Word-Embedding  <cit.>

    ι: 𝒲∪{∅}→ℝ^N.

Then a point 𝐯 of ℝ^N defines a distribution on 𝒲∪{∅} via 

    P_𝐯(w) = exp(-β⟨𝐯, ι(w)⟩)/∑_w' ∈𝒲∪{∅}exp(-β⟨𝐯, ι(w')⟩),

where β is the inverse of temperature. The Moduli-Embedding is a map

    Φ: ℳ→ℝ^N.

Given a sentence 𝐬∈𝒮, Ψ∘ρ(𝐬) is a vector of ℝ^N, which gives the distribution of the next word of 𝐬 via Eq. (<ref>), i.e.,

    P(w|𝐬) = exp(-β⟨Ψ∘ρ(𝐬), ι(w)⟩)/∑_w' ∈𝒲∪{∅}exp(-β⟨Ψ∘ρ(𝐬), ι(w')⟩).




For every n ≥ 1, there exists a map Λ_n: (ℝ^N)^n →ℝ^N that makes the following diagram commutative up to a small error

    𝒮_n = 𝒲^n @>ι^n>> (ℝ^N)^n
    
    @VVρV @VVΛ_nV
    ℳ @>Ψ>> ℝ^N
    .


Namely, given a sentence 𝐬=w_1⋯ w_n, the distribution of the word after 𝐬 is given by the vector Λ_n(ι(w_1),⋯,ι(w_n)) via Eq. (<ref>). In practice, both ι and Λ_n are determined by data and machine learning <cit.>. Currently, the most popular method to construct the map Λ_n is based on the transformer architecture <cit.>.





 §.§ The Fundamental Thermodynamic Relation of LLMs


Given a geometrization (ℝ^N, ⟨·,·⟩, ι), the properties of w ∈𝒲∪{∅} with respect to this geometrization are determined by the embedding vector ϕ(w) ∈ℝ^N. Borrowing language from physics <cit.>, the space ℝ^N is also called the phase space of a language model. Just as in CLM, in a PLM we also have the entropy function S, the internal energy U and Helmholtz free energy F defined on 𝒮.



Suppose we are given a geometrization (ℝ^N, ⟨·,·⟩, ι) and the maps {Λ_n| n≥ 1 } associated with a PLM. The finite set 𝒮_n is viewed as a subspace of (ℝ^N)^n via the embedding ι^n. Then all thermodynamic functions such as S, U and F defined over 𝒮_n admit smooth extensions to functions over (ℝ^N)^n up to small errors.

Let us now look at the entropy function S as an example. This conjecture implies that there exists a differentiable function S: (ℝ^N)^n →ℝ that makes the following diagram commutative up to a small error

    𝒮_n = 𝒲^n @>ι^n>> (ℝ^N)^n
    
    @VVρV @VVSV
    ℳ @>S>> ℝ.




In fact, from the construction of LLMs using the architecture of transformer, the extensions of thermodynamic functions always exist, but the error cannot be guaranteed to be small.


Suppose the coordinate of (ℝ^N)^n is x_i,j with 1 ≤ i ≤ N and 1 ≤ j ≤ n, then we have the fundamental thermodynamic relation of LLMs

    dU = 1/β dS -∑_i,jf_i,jdx_i,j,

where f_i,j is the generalized force given by

    f_i,j=∂S/∂ x_i,j.




As a function on (ℝ^N)^n, we conjecture that the entropy function S for a LLM is convex. 

Suppose 0 ≤λ≤ 1, given two points 𝐱=(x_i,j) and 𝐱'=(x'_i,j) of (ℝ^N)^n, we have

    S(λ𝐱 + (1 - λ) 𝐱' ) ≥λS( 𝐱) + (1 - λ ) S(𝐱' ).


It has an intuitive interpretation, if you mix the two vectors 𝐱=(x_i,j) and 𝐱'=(x'_i,j), the entropy is always greater than the average. Namely, if you mixed the meanings, it always becomes more vague. 








§ CONCLUSION AND OPEN PROBLEMS
 

In this paper, we have used set theory, analysis and probability theory to rigorously give the definitions of Causal Language Model (CLM) and Predicative Language Model (PLM). We propose a generalization of the distributional hypothesis. We introduce the concept of the moduli space of distributions for a CLM, which is the key metric space to understand the power and properties of a language model. Then we define the entropy function for a CLM, study its properties, and show how its zero points are crucial for a language model to incorporate information and logic. Using the entropy function, we formulate an extension conjecture about AGI.

We have used thermodynamics to show that language models admit thermodynamic interpretations. Namely, a sentence is a microstate and a prompt gives us a statistical ensemble, which is distributed according to the Boltzmann distributions. We also formulate the concept of the geometrization of language models and define the Boltzmann manifold. We show the embedding of the moduli space into a Boltzmann manifold determines how accurate can a geometrization be. At last, we discuss how the current LLMs can be viewed as special cases of geometrization with respect to the linear space together with its inner product.

There are many interesting open questions that deserve further study. Perhaps the most straightforward question is what is the best Boltzmann manifold for the geometrization of a language model. Current LLMs all uses the linear space together with its inner product. But it is likely that different languages would need different Boltzmann manifold for the geometrization. For example, some of the possible choices of Boltzmann manifold with non-trivial topology can be

    ℝ^N × S^M, ℝ^N × (S^1)^M,

where S^k is a k-dimensional sphere. For such manifold, it is very interesting to see whether the non-trivial topology has any effect on the properties of the resulting LLM.

Another important question is whether results in modern physics and mathematics can be applied to study language models. For example, whether statistical physics can help us understand the emergent abilities of LLMs. Whether the results in statistical mechanics and differential geometry can offer a better architecture than the transformer. There are numerous results in the study of many-body interaction systems, it is interesting to see whether these results can have applications to LLMs.







plain





 Paper Title: Dirac eigenvalues and the hyperspherical radius.




 

Universität Potsdam, Institut für Mathematik, 14476 Potsdam, Germany
mailto:christian.baer@uni-potsdam.dechristian.baer@uni-potsdam.de
<https://www.math.uni-potsdam.de/baer/>

§ ABSTRACT  
For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved.
When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences.
Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler.
New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler-Einstein manifolds, quaternionic Kähler manifolds and manifolds with a harmonic 1-form of constant length.




[2020]Primary: 53C21, 53C24, 53C27; Secondary: 53C18, 53C26, 53C55, 58J20



Dirac eigenvalues and the hyperspherical radius
    Christian Bär
    August 5, 2024
===============================================





§ INTRODUCTION
 

The idea of hyperspherical radius of a closed connected orientable manifold M goes back to Gromov and Lawson in <cit.> where it was used to prove Geroch's conjecture on the impossibility of positive scalar curvature on tori.
It is defined as the largest number R>0 such that M still dominates the round sphere (R) of radius R in the sense that there exists a 1-Lipschitz map M→(R).
Here n=(M).

If, in addition, M carries a spin structure, then the Dirac operator  is defined on M.
It is elliptic and selfadjoint and hence has discrete real spectrum.
We denote the smallest eigenvalue of ^2 by λ_1(^2).
The main result of the present paper is a bound for λ_1(^2) in terms of the hyperspherical radius.


Let M be a connected closed Riemannian spin manifold of dimension n≥2.
Denote the Dirac operator acting on spinor fields of M by .
Then

    λ_1(^2) ≤n^2/4 (M)^2.

Equality holds in (<ref>) if and only if M is isometric to (R) with R=(M).


In general, the hyperspherical radius is hard to control.
This can even be made precise in complexity theoretical terms, see <cit.>.
The main theorem has a number of applications which are obtained by combining it with known lower bounds for λ_1(^2), which results in upper bounds for the hyperspherical radius.
Amongst these applications are some classically known theorems such as Llarull's scalar curvature rigidity of the standard metric of , the already mentioned Geroch's conjecture and an estimate for the mean curvature of spin fill-ins with nonnegative scalar curvature due to Gromov <cit.>, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler <cit.>.
New applications are concerned with an estimate between the hyperspherical radius and the Yamabe constant as well as improved estimates for the hyperspherical radius on manifolds the special structures, more precisely, on Kähler manifolds, Kähler-Einstein manifolds, quaternionic-Kähler manifolds and manifolds admitting a nontrivial harmonic 1-form of constant length. 

The paper is organized as follows.
In Section <ref> we recall the definition of the hyperspherical radius and discuss a number of its properties which follow directly from the definition.
In Section <ref> we discuss the applications of the main theorem.
The next two sections are devoted to the proof of the main theorem.
In Section <ref> we provide some necessary technical tools and in Section <ref> we carry out the proof.
One is not necessarily restricted to compare the manifold M with round spheres.
In Section <ref> we describe a class of admissible comparison spaces in even dimensions for which an analogue to the main theorem can still be proved (Theorem <ref>).
This class includes the sphere with a metric originating from boundaries of convex domains in Euclidean space.
The odd-dimensional case is left open in this more general setup.
We conclude the paper with a number of remarks and open questions in Section <ref>.



Acknowledgments: 
The author would like to thank Bernhard Hanke, Lennart Ronge and Rudolf Zeidler for many insightful discussions and “SPP2026 - Geometry at Infinity” funded by Deutsche Forschungsgemeinschaft for financial support.



§ THE HYPERSPHERICAL RADIUS




Let M be a closed connected n-dimensional Riemannian manifold.
The hyperspherical radius (M) of M is the supremum of all numbers R>0 such that there exists a Lipschitz map f M→ S^n with Lipschitz constant (f)≤ 1/R and (f)0.


Equivalently, (M) is the supremum of all numbers R>0 such that there exists a 1-Lipschitz map f M→(R) with nonzero degree.
Here (R) denotes the round sphere of radius R.


The supremum in Definition <ref> is a maximum.
In other words, there exists a Lipschitz map f M→ with (f)= 1/(M) and (f)≠0.

Indeed, let R_i↗(M) and let f_i M→ be Lipschitz maps with (f_i)≤ 1/R_i and (f_i)≠0.
By the Arzela-Ascoli theorem, f_i subconverges uniformly to a Lipschitz map f with (f)≤ 1/(M).
Since the (absolute value of) the degree of f_i is at most (f_i)^n, the degrees of the f_i are bounded.
Thus, after passing to a subsequence once more, all f_i have the same nontrivial degree.
The limit map then also has this degree (f)0.
Since there can be no Lipschitz map g M→ with (g)<1/(M) and (g)≠0, we conclude:
there exists a Lipschitz map f M→ with (f)=1/(M) and (f)≠0.



If f M→ is a Lipschitz map with (f)<1/R for some R<(M), then we can approximate f by a smooth map f̃ M→ such that (f̃)<1/R and (f̃)=(f).
Therefore we can can equivalently define the hyperspherical radius as the supremum of all R>0 for which there exists a smooth map f M→ S^n with (f)≤ 1/R and (f)0.
Then, however, the supremum may no longer be a maximum.



If f M→ S^n is a Lipschitz map with (f)≤ 1/R, then the image of a ball B(p,r)⊂ M under f must lie in the ball B(f(p),r/R).
Thus, if (M)/R< (S^n) = π, then f is not surjective and hence has degree 0.
This shows 

    (M)≤(M)/π.




Let f M→ be Lipschitz with (f)=1/R and (f)≠0 where R=(M).
Without loss of generality assume (f)>0.
Denote the volume forms of M and  by _M and _, respectively.
At points where f is differentiable we have f^*_ = ±μ_1⋯μ_n·_M where μ_1,…,μ_n are the singular values of df and the sign is positive if and only if df preserves the orientation.
From μ_j≤ 1/R we find

    (M) 
       =
    ∫_M _M ≥
    R^n ∫_M μ_1⋯μ_n _M 
       ≥ 
    R^n ∫_M f^*_ 
    = 
    R^n (f) ∫__
       ≥ 
    R^n ().

This implies

    (M)^n ≤(M)/().

If we have equality in (<ref>), then we have equality in (<ref>)–(<ref>).
Equality in (<ref>) implies that μ_1=…=μ_n=R almost everywhere, i.e. df is R times a linear isometry at almost all points of M.
Equality in (<ref>) implies that df preserves the orientation almost everywhere. 
Theorem 1 in <cit.> now implies that f M→ S^n is a smooth Riemannian covering (up to rescaling).
By equality in (<ref>), f has degree 1.
Hence, f is an isometry, after rescaling the metric of M.
We have seen:

Equality holds in (<ref>) if and only if M is isometric to (R).




The hyperspherical radius behaves like a radius under rescaling of the metric.
Namely, for any constant λ>0 we have

    (M,λ^2g) = λ·(M,g).




Since f=𝕀→ has degree 1 and Lipschitz constant (𝕀)=1, we have ()≥1.
By Remark <ref>, we also have the inverse inequality.
Thus ()=1.
Remark <ref> now implies

    ((R)) = R.




Let M be a connected closed orientable manifold and let g_1 and g_0 be two Riemannian metrics on M.
Then, if g_1≥ g_0 in the sense of bilinear forms, we have for any Lipschitz map (f (M,g_1)→)≤(f (M,g_0)→).
Hence 

    (M,g_1)≥(M,g_0).




Let (M,g) be a connected closed oriented Riemannian, let p∈ M and let 0<ρ≤(p) where (p) is the injectivity radius of p in M.
We assume that the sectional curvature of M satisfies ≤ρ^2/π^2 on the open ball B(p,ρ).
We rescale the metric g to g̃=(π/ρ)^2g.
With respect to this metric, we have B̃(p,π)=B(p,ρ) and the sectional curvature satisfies ≤ 1 on B̃(p,π).

Pick any point q∈ and a linear isometry A T_pM→ T_q (w.r.t. the metric g̃).
We define a map f (M,g̃)→ by

    f(x):=
    exp_q^∘ A∘ (exp_p^M)^-1(x),     if x∈B̃(p,π),
    
    -q,     else.

Jacobi field estimates together with the sectional curvature bound imply that f is 1-Lipschitz.
Therefore (M,g̃)≥1.
Remark <ref> implies 

    (M,g) ≥ρ/π .

In particular, if we find a ball in M on which the Riemannian exponential map is invertible and the sectional curvature satisfies ≤0, then the radius of this ball provides a lower bound for the hyperspherical radius of M.

















§ APPLICATIONS OF THE MAIN THEOREM

 

Combining the main theorem with known lower bounds for the first Dirac eigenvalue leads to a number of interesting consequences some of which are known theorems.



 §.§ Llarull's theorem


Friedrich proved in <cit.> that if a closed Riemannian spin manifold has scalar curvature ≥ s_0>0 then 

    λ_1(^2) ≥1/4n/n-1s_0 .

Combining this with the main theorem yields

    (M)^2 ≤n(n-1)/s_0.

Thus if s_0≥ n(n-1), then (M)≤1.
In other words, if the scalar curvature of M is at least that of the standard sphere, then all Lipschitz maps f M→ S^n(1) of nontrivial degree must satisfy (f)≥1.
Moreover, if (f)=1, we must have equality in the main theorem and hence f M→ S^n(1) is an isometry.
This proves:


Let M be a closed connected Riemannian spin manifold of dimension n≥2.
Suppose that the scalar curvature of M satisfies ≥ n(n-1).

Then each 1-Lipschitz map f M→ S^n(1) with nonzero degree is a Riemannian isometry.




 §.§ Geroch's conjecture

We follow the conventions of <cit.> and call a connected closed differentiable manifold M enlargeable if for some Riemannian metric g on M and for every R>0 there exists a finite covering πM̂→ M where M̂ is spin and (M̂,π^*g)≥ R.
Since any two Riemannian metrics on M can be estimated by each other, we could equivalently demand the existence of the coverings for all Riemannian metrics g on M.


An enlargeable closed manifold does not admit a Riemannian metric of positive scalar curvature.



Assume that M is enlargeable and has a Riemannian metric g with positive scalar curvature.
By compactness, there exists a positive constant s_0 such that _g≥ s_0.
Then for any spin covering πM̂→ M we also have _π^*g≥ s_0.
Estimate (<ref>) yields (M̂,π^*g)≤n(n-1)/s_0, contradicting enlargeability.


The simplest examples of enlargeable manifolds are tori.
To see this, equip the torus with a flat metric, pass to coverings by larger tori and employ (<ref>).
The nonexistence of positive scalar curvature metrics on tori answers a question attributed to Geroch.



 §.§ Mean curvature of spin fill-ins

Let M be an n-dimensional closed Riemannian spin manifold.
Let h M→ be a smooth function.


A spin fill-in of (M,h) is a pair (X,Φ) where X an (n+1)-dimensional compact connected Riemannian spin manifold with boundary and Φ M→ is a spin-structure preserving isometry such that h=H_X∘Φ.
Here  carries the metric and spin structure induced by those of X and H_X is the unnormalized mean curvature of  in X.
The sign convention is such that the boundary of an (n+1)-dimensional Euclidean ball of radius R has mean curvature n/R. 

If the scalar curvature of X is nonnegative, then we call (X,Φ) an NNSC spin fill-in of (M,h).


It was shown by Hijazi, Montiel, and Zhang in Theorem 6 of <cit.> that if (M,h) admits an NNSC spin fill-in then

    √(λ_1(^2))≥12 min_M h.


Combining this with the main theorem yields

    min_M h ≤n/(M) .


If we have equality in this estimate we must have equality in the main theorem, hence M is isometric to a round sphere.
Moreover, we must have equality in (<ref>). 
By the equality part of Theorem 6 in <cit.>, any NNSC spin fill-in X must be Ricci flat and h must be constant (equal to n/2(M)).

Theorems 4.1 and 4.2 in <cit.> by Shi and Tam then imply that X is isometric to a Euclidean domain bounding a round sphere, hence to a Euclidean ball. 
We summarize:


Let M be an n-dimensional closed Riemannian spin manifold and let h M→ be a smooth function.
If (M,h) admits an NNSC spin fill-in then

    min_M h ≤n/(M) .

If equality holds in (<ref>), then any NNSC spin fill-in is isometric to a Euclidean ball. 


Inequality (<ref>) can be found in Section 4.3 of Gromov's “Four lectures on scalar curvature” <cit.>.
The equality discussion has recently been carried out by Cecchini, Hirsch and Zeidler using different arguments in <cit.>.

The remaining applications are new to the best of our knowledge.




 §.§ A conformal bound

The positive lower bound on scalar curvature can be replaced by a positivity assumption on the Yamabe constant.


Let (M,g) be a closed connected Riemannian spin manifold of dimension n≥3.
Suppose that the Yamabe constant (M,[g]) of the conformal class of g is positive.
Then

    (M)^2 ≤n(n-1)(M,g)^2/n/(M,[g]) .

Equality holds if and only if (M,g) is isometric to a round sphere.



Hijazi has shown in <cit.> that 

    λ_1(^2) ≥n/4(n-1)(M,[g])/(M,g)^2/n .

Combining this with the main theorem yields (<ref>).

If we have equality in (<ref>), then we have equality in the main theorem and hence (M,g) is isometric to a round sphere.

Conversely, if (M,g) is isometric to a round sphere, we can arrange by rescaling that M=.
Then (M)=1 and (M)=n(n-1)(M)^2/n.
Thus we have equality in (<ref>).






















 §.§ Manifolds with special structures

Suitable additional conditions on M can lead to improvements in (<ref>) and hence in (<ref>).
We present three instances of this situation.
We start with Kähler manifolds.


Let M be a closed connected Kähler spin manifold of complex dimension m≥2.
Assume its scalar curvature satisfies ≥ s_0 for some positive constant s_0.
Then

    (M)^2 <
    m^3/m+14/s_0,  if m if odd,
    
    m(m-1)4/s_0,  if m if even,
    m^3/m+24/s_0,  if m≥4 is even and M is Kähler-Einstein.




Kirchberg showed in <cit.>, <cit.> and in <cit.> that 

    λ_1(^2) ≥m+1/4ms_0,  if m if odd,
    m/4(m-1)s_0,  if m if even,
    m+2/4ms_0,  if m≥4 is even and M is Kähler-Einstein.

Combining this with the main theorem yields the weak version of the inequality in the statement.
Equality cannot occur because M would then have to be isometric to S^n which is not a Kähler manifold.
Therefore, the inequalities are strict.


Next we consider quaternionic-Kähler manifolds.
Replacing Kirchberg's estimates by Kramer, Semmelmann and Weingart's estimate (<cit.>)

    λ_1(^2) ≥n+12/4(n+8)s_0

the same reasoning yields:

simple

Let M be a closed connected quaternionic-Kähler spin manifold of real dimension n≥8.
Assume its scalar curvature satisfies ≥ s_0 for some positive constant s_0.
Then

    (M)^2 < n^2(n+8)/n+121/s_0.


default

The existence of a nontrivial harmonic 1-form of constant length also leads to an improvement.
Moroianu and Ornea showed in <cit.> that in this case Friedrich's estimate (<ref>) can be improved to 

    λ_1(^2) ≥1/4n-1/n-2s_0 .

Equality cannot occur in the main theorem because S^n does not admit nontrivial harmonic 1-forms.
This proves:

simple

Let M be a closed connected Riemannian spin manifold of dimension n≥3 which admits a nontrivial harmonic 1-form of constant length.
Suppose that the scalar curvature of M satisfies ≥ s_0 for some positive constant s_0.
Then

    (M)^2 < n^2(n-2)/n-11/s_0.


default





§ SPINORS WITH LIPSCHITZ TWISTS



We now turn to the proof of the main theorem.
The current section provides some preparation for the proof which is then carried out in the next section.

For a linear map A between finite-dimensional Euclidean vector spaces we denote by |A|_1 its trace norm, by |A|_2 its Frobenius norm and by |A|_∞ its spectral norm.
In other words, if 0≤μ_1≤μ_2≤…≤μ_n are the singular values of A, repeated according to multiplicity, then |A|_1=μ_1 + … + μ_n, |A|_2=√(μ_1^2 + … + μ_n^2), and |A|_∞=μ_n.

Throughout this section, let M be a closed Riemannian spin manifold of dimension n≥2.
Let f M→ S^n be a Lipschitz map.
We denote the spinor bundle of M by Σ M and the one of S^n by Σ S^n.
The spinor bundles are Hermitian vector bundles of rank 2^⌊ n/2⌋.
Let  be the connection on Σ S^n induced by the Levi-Civita connection.
For s∈ define the connection

    _Xψ := _Xψ + s X·ψ

on Σ S^n.
Here X·ψ denotes the Clifford product of the tangent vector X with the spinor ψ.
All connections  are metric with respect to the natural Hermitian scalar product on Σ S^n.
Spinors which are parallel w.r.t.  are known as Killing spinors.
Nontrivial Killing spinors exist on S^n for s=1/2 and s=-1/2.
In these cases they trivialize the spinor bundle.
More precisely, there exist Killing spinors for s=1/2 which form an orthonormal basis of Σ S^n at every point of S^n and similarly for s=-1/2.

We pull back Σ S^n along f to M and obtain a Lipschitz bundle f^*Σ S^n, equipped with a Hermitian metric.
For each s∈ we pull back  and obtain a metric connection f^* on f^*Σ S^n.
The corresponding twisted Dirac operator _f,s is defined in the usual manner and yields a selfadjoint operator H^1(M;Σ M⊗ f^*Σ S^n) → L^2(M;Σ M⊗ f^*Σ S^n).
Here L^2(M;Σ M⊗ f^*Σ S^n) and H^1(M;Σ M⊗ f^*Σ S^n) are defined as the completions of the space of all Lipschitz sections of Σ M⊗ f^*Σ S^n with respect to the usual L^2-norm ·_L^2 and Sobolev norm ·_L^2+∇·_L^2, respectively.
See Section 4 of <cit.> for further details, although many complications treated in <cit.> do not arise in our setup because all our Riemannian metrics are smooth.

We denote by  the trivial ^-bundle over M with standard Hermitian scalar product and standard connection d.
The Dirac operators on Σ M and on Σ M⊗ will both be denoted by .
The have the same spectrum, up to multiplicity.

We choose ∇^1/2-parallel spinor fields (ψ_α^+)_α=1,…, which form an orthonormal basis of Σ S^n at each point of S^n.
We pull back the ψ_α^+ along f and obtain sections Ψ_α^+:=f^*ψ_α^+ of f^*Σ S^n which form an orthonormal basis at each point of M and satisfy (f^*)_XΨ_α^+ = (s-1/2) f^*(df(X)·ψ_α^+).

Each ϕ∈Σ M⊗ f^*Σ S^n can be expanded uniquely as

    ϕ = ∑_αϕ_α⊗Ψ_α^+ .

The map ϕ↦ (ϕ_α)_α=1,…,2^⌊n/2⌋ defines a unitary vector bundle isomorphism 

    U_+Σ M⊗ f^*Σ S^n →Σ M⊗

of Lipschitz regularity.
We have ϕ_L^2^2 = U_+ϕ_L^2^2 = ∑_αϕ_α_L^2^2.
Moreover, ϕ∈ H^1(M;Σ M⊗ f^*Σ S^n) if and only if U_+ϕ∈ H^1(M;Σ M⊗)  which is equivalent to all ϕ_α∈ H^1(M;Σ M).

Similarly, we can use an orthonormal basis of ∇^-1/2-parallel spinor fields (ψ_α^-)_α=1,…,.
Replacing Ψ_α^+ in (<ref>) by Ψ_α^- := f^*ϕ_α^- we obtain a second unitary vector bundle isomorphism U_-Σ M⊗ f^*Σ S^n →Σ M⊗ with similar properties as U_+.

For α,β=1,…, let ζ_αβ^± be the complex 1-forms defined by

    ζ_αβ^±(X) =  df(X)·ψ_α^±,ψ_β^± .

Since f is Lipschitz, the differential df exists almost everywhere on M and defines an L^∞-section.
Hence ζ_αβ^± is an L^∞-1-form on M.
Let Z_αβ^± be the corresponding L^∞-vector fields on M, i.e.,

    Z_αβ ^±
    = 
    ∑_j=1^n ζ_αβ^±(e_j)e_j
    =
    ∑_j=1^n  df(e_j)·ψ_α^±,ψ_β^±e_j,

where e_1,…,e_n is a local orthonormal tangent frame.
We obtain two L^∞-endomorphism fields Z^+ and Z^- on Σ M⊗ by

    Z^± (ϕ_α)_α↦(∑_β Z_βα^±·ϕ_β)_α .



We have 

    U_+ ∘_f,s∘ U_+^-1 =  + (s-12)Z^+

and 

    U_- ∘_f,s∘ U_-^-1 =  + (s+12)Z^- .




We compute, using a local orthonormal tangent frame e_1,…,e_n:

    _f,sϕ   =
    _f,s∑_αϕ_α⊗Ψ_α^±
       =
    ∑_αϕ_α⊗Ψ_α^± + ∑_j,α e_j·ϕ_α⊗_e_jΨ_α^±
       =
    ∑_αϕ_α⊗Ψ_α^± + ∑_j,α e_j·ϕ_α⊗ (s∓12)f^*(df(e_j)·ψ_α^±) 
       =
    ∑_αϕ_α⊗Ψ_α^± + (s∓12)∑_j,α,β e_j·ϕ_α⊗f^*(df(e_j)·ψ_α^±),Ψ_β^±Ψ_β^±
       =
    ∑_αϕ_α⊗Ψ_α^± + (s∓12)∑_j,α,β e_j·ϕ_α⊗df(e_j)·ψ_α^±,ψ_β^±Ψ_β^±
       =
    ∑_α(ϕ_α + (s∓12)∑_j,βdf(e_j)·ψ_β^±,ψ_α^±e_j·ϕ_β)⊗Ψ_α^±
       =
    ∑_α(ϕ_α + (s∓12)∑_β Z_βα^±·ϕ_β)⊗Ψ_α^± .

This proves the lemma.


The lemma implies that =U(+Z^-)U^-1 where U=U_+U_-^-1.
In particular,  and +Z^- have the same spectrum.
Similarly,  and -Z^+ are conjugate and hence have the same spectrum.































For v,w∈ T_pM we consider the endomorphism 

    τ_v,w:=v· w⊗ d_pf(v)· d_pf(w)

of (Σ M⊗ f^*Σ S^n)_p.
The following lemma provides the crucial estimate:


Let s∈ and let ϕ∈ H^1(M;Σ M⊗ f^*Σ S^n) with _f,sϕ=0.
We write U_+ϕ =: (ϕ_α)_α if s≥0 and U_-ϕ =: (ϕ_α)_α if s<0.
Then

    ∑_α |ϕ_α|^2
    ≤(12-|s|)^2  |df|_1^2  ∑_α |ϕ_α|^2

holds almost everywhere on M.

Moreover, if |s|≠1/2, if f is differentiable at p∈ M, if d_pf is a linear isometry and if equality holds in (<ref>) at p, then

    τ_v,w ϕ(p) = ϕ(p)

for all orthonormal v,w∈ T_pM.



First note that since ϕ_α and Dϕ_α are L^2-spinors and |df|_1 is an L^∞-function, both sides in (<ref>) are L^1-functions.
Hence it makes sense to evaluate them almost everywhere.

Now we fix a point p∈ M at which the Lipschitz map f is differentiable and choose orthonormal bases e_1,…,e_n of T_pM and e_1^0,…,e_n^0 of T_f(p)S^n such that df(e_j)=μ_j e_j^0 where 0≤μ_1≤…≤μ_n are the singular values of df at p.

W.l.o.g. we assume s≥0, the case s<0 being entirely analogous.
From Lemma <ref> we have (ϕ_α)_α = (1/2-s)Z^+(ϕ_α)_α.
We compute at p:

    ∑_α |ϕ_α|^2
       =
    (12-s)^2|Z^+(ϕ_α)_α|^2 
       =
    (12-s)^2∑_α∑_β Z_βα^+ϕ_β, ∑_γ Z_γα^+ϕ_γ
       =
    (12-s)^2∑_α,β,γ,δ∑_β Z_βα^+ϕ_β, ∑_γ Z_γδ^+ϕ_γψ_α^+,ψ_δ^+
       =
    (12-s)^2∑_α,β,γ,δ,j,kdf(e_j)·Ψ_β^+,Ψ_α^+e_j·ϕ_β , df(e_k)·Ψ_γ^+,Ψ_δ^+e_k·ϕ_γψ_α^+,ψ_δ^+
       =
    (12-s)^2∑_α,β,γ,δ,j,k e_j·ϕ_β , e_k·ϕ_γdf(e_j)·Ψ_β^+,Ψ_α^+ψ_α^+,df(e_k)·Ψ_γ^+,Ψ_δ^+ψ_δ^+
       =
    (12-s)^2∑_β,γ,j,k e_j·ϕ_β , e_k·ϕ_γdf(e_j)·Ψ_β^+,df(e_k)·Ψ_γ^+
       =
    (12-s)^2∑_β,γ,j,kμ_jμ_k e_j·ϕ_β , e_k·ϕ_γe^0_j·Ψ_β^+,e^0_k·Ψ_γ^+
       =
    (12-s)^2∑_β,γ,j,kμ_jμ_k e_j· e_k·ϕ_β , ϕ_γe^0_j· e^0_k·Ψ_β^+,Ψ_γ^+
       =
    (12-s)^2∑_j,kμ_jμ_k (e_j· e_k⊗ e^0_j· e^0_k)ϕ,ϕ .

We investigate the summands of the right-hand side for fixed j and k.
If j=k, then we have

    (e_j· e_k⊗ e^0_j· e^0_k)ϕ,ϕ
    =
    ((-1)⊗ (-1))ϕ,ϕ
    =
    |ϕ|^2.

If j≠ k, then e_j· e_k⊗ e^0_j· e^0_k is a selfadjoint involution.
Therefore,

    (e_j· e_k⊗ e^0_j· e^0_k)ϕ,ϕ≤|ϕ|^2

with equality if and only if 

    (e_j· e_k⊗ e^0_j· e^0_k)ϕ=ϕ .

Inserting (<ref>) and (<ref>) into (<ref>) yields

    ∑_α |ϕ_α|^2
    ≤(12-s)^2∑_j,kμ_jμ_k  |ϕ|^2
    =
    (12-s)^2 |df|_1^2 ∑_α |ϕ_α|^2 .

If d_pf is an isometry, then we have μ_1=…=μ_n=1 at p and the orthonormal basis e_1,…,e_n can be chosen arbitrarily.
Together with (<ref>) this implies (<ref>).


Finally, we will need the following lemma:


We have 

    U_+^-1Z^+U_+ = U_-^-1Z^- U_- = ∑_j e_j ⊗ df(e_j)

where e_1,…,e_n denotes a local orthonormal frame and the tangent vectors act by Clifford multiplication.



Writing U_±ϕ = (ϕ_α)_α we compute:

    U_±^-1 Z^± U_±ϕ   =
    U_±^-1 Z^± (ϕ_α)_α
       =
    ∑_αβ Z_βα^±·ϕ_β⊗Ψ_α^±
       =
    ∑_αβ jdf(e_j)·ψ_β^±,ψ_α^±e_j ·ϕ_β⊗Ψ_α^±
       =
    ∑_β j e_j ·ϕ_β⊗ df(e_j)·Ψ_β^±
       =
    ∑_j (e_j ⊗ df(e_j))ϕ .






§ PROOF OF THE MAIN THEOREM
 
 

In this section we prove the main theorem.
Let f M→ S^n be a Lipschitz map with (f)0 and (f)=1/(M).



 §.§ Proof of (_f,0)0 if the dimension n is even.


In even dimension we have the chirality splittings of Σ M=Σ^+M⊕Σ^-M and Σ S^n=Σ^+S^n⊕Σ^-S^n at our disposal.
The connection  preserves the splitting Σ S^n=Σ^+S^n⊕Σ^-S^n (unlike  for s0).
Thus the twisted Dirac operator _f,0 can be restricted to four operators:

    _f,0^++    C^∞(M;Σ^+ M⊗ f^*Σ^+ S^n)→ C^∞(M;Σ^- M⊗ f^*Σ^+ S^n),
    _f,0^+-    C^∞(M;Σ^+ M⊗ f^*Σ^- S^n)→ C^∞(M;Σ^- M⊗ f^*Σ^- S^n),
    _f,0^-+    C^∞(M;Σ^- M⊗ f^*Σ^+ S^n)→ C^∞(M;Σ^+ M⊗ f^*Σ^+ S^n),
    _f,0^–    C^∞(M;Σ^- M⊗ f^*Σ^- S^n)→ C^∞(M;Σ^+ M⊗ f^*Σ^- S^n).

The Atiyah-Singer index theorem applies (see Theorem 4.8 in <cit.>) and yields

    (_f,0^++) + (_f,0^–)
       =
    (_f,0^++) - (_f,0^+-) 
       =
    (M)∪ f^*((Σ^+ S^n) - (Σ^- S^n)),[M]
       =
    (M)∪ (-1)^n/2f^*(TS^n),[M]
       =
    (-1)^n/2 f^*(TS^n) ,[M]
       =
    (-1)^n/2(f)(TS^n),[S^n]
       = 
    (-1)^n/2· 2·(f).

Here  denotes the -class,  the Chern character and  the Euler class.
In the last line we used once more that n is even and hence S^n has Euler characteristic 2.
If (-1)^n/2·(f)>0 we find that (_f,0^++)>0 or (_f,0^–)>0.
If (-1)^n/2·(f)<0 we find that (_f,0^++)<0 or (_f,0^–)<0 and hence (_f,0^-+)>0 or (_f,0^+-)>0.
In any case, there exists a nontrivial ϕ∈ H^1(M;Σ M⊗ f^*Σ S^n) with _f,0ϕ=0.
We keep in mind, however, that ϕ is a section of one of the subbundles Σ^± M⊗ f^*Σ^± S^n.



 §.§ Proof of (_f,s)0 for some s∈[-1/2,1/2] if the dimension n is odd.

Denote the spectral flow of the operator family [-1/2,1/2]∋ s↦_f,s by (_f,·).
By Lemma <ref>, this is the same as the spectral flow of the family [0,1]∋ s↦ + sZ^-.

For smooth f, this spectral flow has been computed to |(_f,·)|=|(f)| in <cit.>[Our connection  corresponds to the connection ∇_s+1/2=d+(s+1/2)ω in <cit.>.] based on <cit.>.
To see that the result remains true for our Lipschitz map f, choose a smooth map f_1 M→ S^n such that (f(x),f_1(x))<π for all x∈ M.
Then we obtain a homotopy [0,1]∋ t↦ f_t between f=f_0 and f_1 by joining the image points by the unique shortest geodesic in  connecting them.
In particular, (f)=(f_1).
By homotopy invariance of the spectral flow, we have

    ([-12,   12]∋ s ↦_f,s) 
       =
    ([0,1]∋ t↦_f_t,-1/2)
    + ([-12,12]∋ s↦_f_1,s)
    - ([0,1]∋ t↦_f_t,1/2) 
       =
    ([-12,12]∋ s↦_f_1,s) .

The negative sign of the last term in (<ref>) comes from the fact that we have to run that operator family backwards.
Moreover, ([0,1]∋ t↦_f_t,-1/2)=([0,1]∋ t↦_f_t,1/2) follows from _f_t,-1/2 and _f_t,1/2 having the same spectrum, cf. Section <ref>.

Hence (_f,·)≠0 and therefore not all _f,s can be invertible. 
Since the _f,s are selfadjoint, at least one _f,s must have a nontrivial kernel where s∈[-1/2,1/2].



 §.§ Proof of estimate (<ref>) in arbitrary dimension n


Now we return to arbitrary dimension n.
Let ϕ∈(_f,s)0 for some s∈[-1/2,1/2].
We put (ϕ_α)_α:=U_+ϕ if s≥0 and (ϕ_α)_α:=U_-ϕ if s<0.
Then we have ϕ_α∈ H^1(M;Σ M), not all ϕ_α are zero, and, by Lemma <ref>,

    ∑_α |ϕ_α|^2
    ≤(12-|s|)^2|df|_1^2  ∑_α |ϕ_α|^2
    ≤14 |df|_1^2  ∑_α |ϕ_α|^2 .

Since (f)=1/(M) all singular values of df are bounded by 1/(M) and hence |df|_1≤ n/(M).
Integrating over M we find

    λ_1(^2)∑_αϕ_α_L^2^2
    ≤∑_αϕ_α_L^2^2
    ≤n^24(M)^2∑_αϕ_α_L^2^2 .

This implies λ_1(^2) ≤n^2/4(M)^2 and concludes the proof of (<ref>).



 §.§ Equality for the sphere.

If M=(R), then the smallest Dirac eigenvalue is known to be λ_1(^2)=n^2/4R^2, see e.g. Theorem 1 in <cit.>.
Thus equality holds in (<ref>) in this case.



 §.§ Beginning of the equality discussion in arbitrary dimension n.

Conversely, assume equality in (<ref>).
For notational convenience, we assume that (M)=1 which can be achieved by rescaling the metric.
Hence, (f)=1 and thus the singular values of df satisfy μ_j≤1 almost everywhere.

Let ϕ=∑_αϕ_α⊗Ψ_α∈ H^1(M;Σ M⊗ f^*Σ S^n) be nontrivial with _f,sϕ=0 with s∈[-1/2,1/2].
Equality in (<ref>) implies equality in (<ref>) and thus ϕ_α are eigenspinors for ^2 for the eigenvalue λ_1(^2).
In particular, they are smooth and the zero locus of the nontrivial ϕ_α have Hausdorff codimension ≥2 by the main theorem in <cit.>.
Thus, ϕ is continuous and its zero locus  has Hausdorff codimension ≥2.
In particular, it is a nullset and its complement is open and connected.

Moreover, equality in (<ref>) implies equality in |df|_1≤ n/(M)=n unless ϕ=0.
Therefore, all μ_j=1.
This means that df is a linear isometry almost everywhere.
It remains to show that d_pf preserves the orientation for almost all p or it reverses the orientation for almost all p since it then follows from Theorem 1 in <cit.> that f M→ S^n is a smooth Riemannian covering.
Since M is connected and S^n is simply connected, f then is an isometry.

Equality in (<ref>) implies equality in (<ref>).
Thus s=0 and, by Lemma <ref>, equation (<ref>) holds almost everywhere.

Let ω_ be the Clifford volume element for Σ M.
It is a field of selfadjoint involutions on Σ M and can locally be written as ω_ = i^⌊n+1/2⌋e_1⋯ e_n where e_1,…,e_n is a positively oriented local orthonormal tangent frame.
Similarly, let ω_^0 be the Clifford volume element for Σ S^n.



 §.§ Conclusion of the equality discussion for even dimension n.


We conclude the equality discussion in (<ref>) if the dimension n is even.
The volume element ω_ acts as ± 1 on Σ^± M and similarly for ω_^0.
Since ϕ is a section of one of the subbundles Σ^± M⊗ f^*Σ^± S^n, we have that 

    (ω_⊗ω_^0)ϕ=ϕ

for a constant sign =± 1.

Let p∈ M such that f is differentiable at p.
Let e_1,…,e_n be a positively oriented orthonormal basis of T_pM.
Since d_pf is a linear isometry, d_pf(e_1),…,d_pf(e_n) is an orthonormal basis of T_f(p)S^n.
It is positively oriented if and only if d_pf is orientation preserving. 
Therefore,

    (-1)^⌊n+1/2⌋ω_⊗ω_^0 
    = 
    σ(p) e_1⋯ e_n⊗ d_pf(e_1) ⊗ d_pf(e_n)
    =
    σ(p) τ_e_1,e_2∘τ_e_3,e_4∘…∘τ_e_n-1,n

where σ(p)=+1 if d_pf is orientation preserving and σ=-1 if d_pf reverses the orientation.
From (<ref>) we find

    (-1)^⌊n+1/2⌋(ω_⊗ω_^0)ϕ(p) = σ(p) ϕ(p).

Comparing (<ref>) and (<ref>) we find σ=(-1)^⌊n+1/2⌋ almost everywhere.
Therefore, d_pf preserves the orientation for almost all p or it reverses the orientation for almost all p.
This concludes the proof for even n.



 §.§ Conclusion of the equality discussion for odd dimension n.

Finally, assume equality in (<ref>) and that the dimension n is odd.
In this case ω_ and ω_C^0 act as the identity on the spinor bundles. 
For p, e_1,…,e_n, and σ as above we get 

    (-1)^⌊n+1/2⌋ϕ(p)
       =
    (-1)^⌊n+1/2⌋(ω_⊗ω_^0) ϕ(p) 
       =
    σ(p)(e_1⊗ d_pf(e_1))τ_e_2,e_3∘…∘τ_e_n-1,nϕ(p)
       =
    σ(p)(e_1⊗ d_pf(e_1))ϕ(p) .

Since e_1 can be chosen arbitrarily, this yields 

    (v⊗ d_pf(v))ϕ(p)
    =
    (-1)^⌊n+1/2⌋σ(p)ϕ(p)

for all unit vectors v∈ T_pM.
Lemma <ref> implies U_+^-1Z^+U_+ϕ = (-1)^⌊n+1/2⌋nσϕ.
From Lemma <ref> we get

    U_+ϕ   =
    U_+_f,0ϕ + 12 Z^+U_+ϕ 
    =
    0 + 12 U_+((-1)^⌊n+1/2⌋nσϕ) 
    =
    (-1)^⌊n+1/2⌋nσ2 U_+ϕ .

Writing U_+ϕ=(ϕ_α)_α this implies for each α:

    ϕ_α = (-1)^⌊n+1/2⌋nσ2ϕ_α .

Therefore,

    ∑_α |ϕ_α|^2
    =
    n^24∑_α |ϕ_α|^2
    =
    λ_1(^2)∑_α |ϕ_α|^2

Integration over M yields

    ∑_αϕ_α_L^2^2
    =
    λ_1(^2)∑_αϕ_α_L^2^2

and hence ^2ϕ_α = λ_1(^2)ϕ_α.
In particular, all ϕ_α are smooth. 
Thus (<ref>) implies that σ is smooth and hence locally constant on M∖.
Since M∖ is connected, σ is constant which concludes the proof.




§ MORE GENERAL TARGET SPACES

We can allow more general target spaces than just the standard sphere .
At least if the dimension n is even (which we assume for the rest of this section), a good class of targets seems to be the following.


Let  be a connected closed n-dimensional Riemannian spin manifold with Euler number χ()0.
Let W be a smooth field of positive semidefinite symmetric endomorphisms of the tangent bundle T.
The pair (,W) is called an admissible comparison space if the spinor bundle Σ of  can be trivialized by spinors ψ satisfying 

    ∇_Xψ = 12 W(X)·ψ

as well as by spinors ψ satisfying 

    ∇_Xψ = -12 W(X)·ψ .



Spinors satisfying (<ref>) or (<ref>) are known as generalized Killing spinors.


The admissible comparison space we have been using throughout this article is (,W)=(,𝕀_T).



More generally, let Ω⊂^n+1 be a convex domain with smooth boundary .
The boundary inherits a Riemannian metric and a spin structure.
Let ν be the exterior unit normal vector field along  and let W(X)=-∇_Xν the Weingarten map.
Then W is a positive semidefinite symmetric endomorphism field.

Since n is even, the restriction of the spinor bundle Σ^n+1 to  can be naturally identified with Σ.
We can trivialize Σ^n+1 or Σ^+^n+1 by parallel spinors.
Restricting a parallel spinor ψ to , the Gauss equation for the spinorial connection (see e.g. Proposition 2.1 in <cit.>) yields

    0 = ∇^^n+1_Xψ 
    = ∇_Xψ + 12 W(X)⋆ν⋆ψ 
    = ∇_Xψ + 12 W(X)·ψ .

Here "⋆" denotes Clifford multiplication w.r.t. ^n+1 and "·" is Clifford multiplication for .
Thus we have a trivialization of Σ by spinors solving (<ref>).
If ψ solves (<ref>), we can put ϕ=ν⋆ψ.
We then get

    ∇_Xϕ   =
    ∇^^n+1_Xϕ - 12 W(X)·ϕ 
    =
    ∇^^n+1_X(ν⋆ψ) - 12 W(X)·ϕ
       =
    ∇_Xν⋆ψ + ν⋆∇^^n+1_Xψ - 12 W(X)·ϕ
    =
    -W(X)⋆ψ + 0 - 12 W(X)·ϕ
       =
    W(X)⋆ν⋆ϕ - 12 W(X)·ϕ
    =
    12 W(X)·ϕ .

Thus we also have a trivialization of Σ by spinors solving (<ref>).
This shows that (,W) is an admissible comparison space.



Let (,W) be an admissible comparison space and M be a closed connected Riemannian manifold with ()=(M)=n.
The hyper--radius (M) of M is the supremum of all numbers R>0 such that there exists a Lipschitz map f M→ with Lipschitz constant (f)≤ 1/R and (f)0.


For = and W=𝕀_T we recover the hyperspherical radius of M.
The same reasoning as in Section <ref> yields:



  * (M)≤(M)/(),

  * (M)^n ≤(M)/()


with equality if and only if M is isometric to  up to rescaling to the metric,

  * (M,λ^2g) = λ·(M,g),

  * 𝖱𝖺𝖽_(,λ^2g_)(M) = 1λ𝖱𝖺𝖽_(,g_)(M) ,

  * ()=1,

  * The supremum in Definition <ref> is a maximum.
In other words, there exists a Lipschitz map f M→ with (f)= 1/(M) and (f)≠0.



Let (,W) be an admissible comparison space and let M be a connected closed Riemannian spin manifold with even dimension (M)=()≥2.
Denote the Dirac operator acting on spinor fields of M by .
Then

    λ_1(^2) ≤max_M (W)^2/4 (M)^2.

If equality holds in (<ref>) and W is positive definite almost everywhere, then (W) is constant and there is a finite Riemannian (up to rescaling the metric) covering map M→.


We first prove an auxiliary lemma.



Let U and V be n-dimensional Euclidean vector spaces, let F U→ V be linear and let W V→ V be symmetric and positive semidefinite.
Then

    |W∘ F|_1 ≤(W) |F|_∞.

If W is positive definite and equality holds in (<ref>), then F is |F|_∞ times an isometry.



If F=0, then inequality (<ref>) is trivial.
So assume F≠ 0.
Let e_1,…,e_n be an orthonormal basis of U and e_1^0,…,e_n^0 one of V such that W(F(e_j))=μ_j e_j^0 where μ_j are the singular values of W∘ F.
Since W is positive semidefinite it has a positive semidefinite symmetric square root L V→ V.
We compute

    |W∘ F|_1 
       =
    ∑_j μ_j 
       =
    ∑_j  W(F(e_j)), e_j^0
       =
    ∑_j  L(F(e_j)), L(e_j^0)
       =
    ∑_jk L(F(e_j)),e_k^0e_k^0, L(e_j^0)
       =
    ∑_jk1/√(|F^*|_∞) L(F(e_j)),e_k^0·√(|F^*|_∞)e_k^0, L(e_j^0)
       ≤12∑_jk( 1/|F^*|_∞ L(F(e_j)),e_k^0^2 + |F^*|_∞e_k^0, L(e_j^0)^2)
       =
    1/2|F^*|_∞∑_jk e_j,F^*(L(e_k^0))^2 + |F^*|_∞/2∑_j|L(e_j^0)|^2
       =
    1/2|F^*|_∞∑_k|F^*(L(e_k^0))|^2 + |F^*|_∞/2|L|_2^2
       ≤|F^*|_∞/2∑_k|L(e_k^0)|^2 + |F^*|_∞/2|L|_2^2
       =
    |F^*|_∞· |L|_2^2
       =
    |F|_∞·(W).

This proves (<ref>).

Now assume that W is positive definite and that equality holds in (<ref>).
Again, we can assume F≠ 0.
Then, in particular, equality must hold in the first inequality of the above computation.
This means that 

    √(|F^*|_∞)e_k^0, L(e_j^0) = 1/√(|F^*|_∞) L(F(e_j)),e_k^0

for all j and k.
Thus 

    |F^*|_∞e_k^0, L(e_j^0) =  L(F(e_j)),e_k^0

for all j and k and hence

    |F^*|_∞ L(e_j^0) = L(F(e_j))

for all j.
Composing both sides with L yields

    |F^*|_∞ W(e_j^0)=W(F(e_j)) = μ_j e_j^0.

Thus e_1^0,…,e_n^0 form an eigenbasis for W for the eigenvalues μ_1/|F^*|_∞,…,μ_n/|F^*|_∞.
Since W is positive definite, all these eigenvalues are positive and we can compute

    F(e_j) 
       = 
    W^-1∘ W∘ F(e_j) 
    = 
    |F^*|_∞/μ_jμ_j e_j^0
    =
    |F^*|_∞ e_j^0
    =
    |F|_∞ e_j^0 .

Therefore all singular values of F are equal to |F|_∞ and hence F is |F|_∞ times an isometry.



We indicate the necessary changes to the proof of the main theorem.
We can now define the unitary vector bundle isomorphisms U_±Σ M⊗ f^*Σ→Σ M⊗ in (<ref>) using the generalized Killing spinors on  instead of the Killing spinors on .
If we replace df by W∘ df in the definition of the 1-forms ζ_αβ^± and in Lemmas <ref>, <ref>, and <ref>, then these lemmas still hold true.

The first step <ref> of the proof of the main theorem still works because we have assumed that  has nontrivial Euler number which insures nontriviality of the index.
Proof step <ref> also works and yields a nontrivial tuple of spinor fields ϕ_α∈ H^1(M;Σ M) satisfying

    ∑_α |ϕ_α|^2
    ≤14 |W∘ df|_1^2  ∑_α |ϕ_α|^2 .

Lemma <ref> implies 

    ∑_α |ϕ_α|^2
    ≤14 (W)^2|df|_∞^2  ∑_α |ϕ_α|^2 
    ≤14 (W)^2(f)^2  ∑_α |ϕ_α|^2 .

This proves (<ref>).






Now assume that equality holds in (<ref>) and that W is almost everywhere positive definite.
Then we have equality in (<ref>) and hence |df|_∞=(f) almost everywhere. 
From the equality case in Lemma <ref> we deduce that df is (f) times a linear isometry almost everywhere.

As in the proof of Lemma <ref> find that (<ref>) holds.
This implies τ_v,w ϕ(p) = (f)^2ϕ(p) for almost all p∈ M and orthonormal tangent vectors v,w∈ T_pM.
Now the argument in Section <ref> applies and shows that f M→ is a Riemannian covering (up to scaling by the factor (f)).
Finally, equality in (<ref>) implies that (W)=max_M(W) is constant. 




Consider the situation described in Example <ref>, i.e., =∂Ω where Ω⊂^n+1 is a convex domain with smooth boundary. 
We assume that the Weingarten map W of the boundary of Ω is positive definite almost everywhere.
If we now have equality in (<ref>), then  has constant mean curvature and must be a round sphere by Alexandrov's theorem.
Since the sphere is simply connected, the Riemannian covering M→ must be an isometry (up to scaling of the metric).
Thus M is a round sphere as well.

As an application of these considerations let h∈ C^∞() be such that min_ h ≥max_(W).
Assume that there is an NNSC spin fill-in of (,h).
As in Section <ref> we find

    max_(W) ≤min_ h ≤ 2√(λ_1(^2))≤max_(W)/() = max_(W).

Thus we have equality in (<ref>) and hence  is a round sphere.
Reasoning as in Section <ref>, we conclude that the spin fill-in is a Euclidean ball.






§ CONCLUDING REMARKS AND QUESTIONS



We conclude with a few remarks and some related questions.



 §.§ Alternative approach to the odd-dimensional case

There is an alternative way to prove the estimate (<ref>) for odd dimension n, completely avoiding the spectral flow argument in Section <ref>.
This alternative argument is more in the spirit of Llarull's way to treat the odd-dimensional case in <cit.>.

Namely, let ρ>0 and put M̃ :=M×𝕊^1(ρ) and where we equip 𝕊^1(ρ) with the spin structure for which it has a parallel spinor.
Then λ_1(^2)=λ_1(^2) where  is the Dirac operator of M̃.

The singular values of the differential of f×𝕀M̃ =M×𝕊^1(ρ) → S^n×𝕊^1(2) are same as those for f plus one singular value 2/ρ coming from the S^1-direction.

There exists a smooth map h S^n× S^1(2)→ S^n+1 with Lipschitz constant 1 and of degree 1, see Lemma 2.3 in <cit.>.
Thus the Lipschitz map F:=h∘(f×𝕀)M̃→ S^n+1 has degree 1 and the singular values of its differential satisfy 0≤μ_j≤1/R for j=1,…,n and μ_n+1≤2/ρ.
The discussion of the even-dimensional case now gives

    ∑_αϕ_α_L^2^2
       ≤14|dF|_1^2 ∑_αϕ_α_L^2^2 
       ≤14(|df|_1+2ρ)^2 ∑_αϕ_α_L^2^2 
       ≤14(n(M)+2ρ)^2 ∑_αϕ_α_L^2^2 .

We find

    λ_1(^2)
    =
    λ_1(^2)
    ≤14(n(M)+2ρ)^2 .

Taking the limit ρ↗∞ concludes the proof of (<ref>) for odd n.


Can the equality discussion be carried out using this approach?




 §.§ A refined hyperspherical radius


In Section <ref> it has been shown that 

    λ_1(^2) ≤14 |df|_1^2

for any Lipschitz map f M→.
Together with |df|_1 ≤ n (f) this yields the estimate (<ref>).

We can introduce a refined hyperspherical radius by letting ^(M) to be the supremum of all numbers R>0 such that there exists a Lipschitz map f M→ with |df|_1≤ n/R and (f)≠0.
Then ^(M)≥(M) and we have actually proved the potentially stronger estimate

    λ_1(^2) ≤n^2/4 ^(M)^2.


What is the equality case in (<ref>)?




 §.§ Λ^2-contracting maps

Another way to modify the hyperspherical radius would be to let ^Λ^2(M) be the supremum of all numbers R>0 such that there exists a Lipschitz map f M→ with |Λ^2df|_∞≤ 1/R^2 and (f)≠0.
Here Λ^2dfΛ^2T_pM →Λ^2 T_f(p) is the induced linear map on 2-vectors and |·|_∞ denotes the spectral norm.
Since |Λ^2df|_∞≤|df|_∞^2, we have ^Λ^2(M)≥(M).


Does the estimate

    λ_1(^2) ≤14 |Λ^2 df|_∞

hold and does equality imply that f is a Riemannian isometry?

If true, we would have

    λ_1(^2) ≤n^2/4 ^Λ^2(M)^2

and the discussion in Section <ref> would yield Theorem C in <cit.> which is stronger than Theorem B.




 §.§ More general target spaces

In Section <ref> we have discussed how one can replace  by more general comparison spaces  in even dimensions.


What would be a good class of admissible comparison spaces in odd dimensions for which Theorem <ref> still holds?






















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 Paper Title: Singularities of generalized timelike minimal surfaces in Lorentz-Minkowski 3-space.



Singularities of timelike minimal surfaces]Singularities of generalized timelike minimal surfaces
 in Lorentz-Minkowski 3-space
S. Akamine]Shintaro Akamine
[Shintaro Akamine]
Department of Liberal Arts, College of Bioresource Sciences,
Nihon University, 
1866 Kameino, Fujisawa, Kanagawa, 252-0880, Japan
akamine.shintaro@nihon-u.ac.jp




[2010]Primary 53A10; Secondary 53B30, 57R45.


§ ABSTRACT 

A timelike minimal surface in Minkowski 3-space is a surface whose induced metric is Lorentzian and with vanishing mean curvature. Such surfaces have many kinds of singularities. In this paper, we prove existence and non-existence theorems of singularities of timelike minimal surfaces, and show that various diffeomorphism types of singularities that do not appear on these Riemannian counter parts, such as minimal surfaces in Euclidean space and maximal surfaces in Minkowski space, appear on timelike minimal surfaces. We also give criteria for cuspidal butterfly, cuspidal S_1 singularity, (2,5)-cuspidal edge, cuspidal beaks and D_4 singularity of timelike minimal surfaces. Finally, duality and invariance theorems for these singularities and examples are given.  


[
    [
    August 5, 2024
==================







§ INTRODUCTION

A spacelike surface in the three-dimensional Lorentz-Minkowski space 𝕃^3 whose mean curvature vanishes identically is called a maximal surface. It is known that only plane is a complete maximal surface <cit.>, and hence maximal surfaces with singular points has been considered as a natural object of study. Here, a singular point or singularity is a point at which the surface is not immersed.
There have been many remarkable results concerning singularities themselves. In particular, diffeomorphism types of various singularities have been determined by using a pair (g, ω) of a meromorphic function g and a holomorphic 1-form ω on a Riemann surface, which is called Weierstrass data and describes a maximal surface. See <cit.> and their references. Moreover, Fujimori, Saji, Umehara and Yamada <cit.> proved that generic singularities on maximal surfaces (more precisely, the class of maximal surfaces with singularities called maxfaces) consist of cuspidal edges, swallowtails and cuspidal cross caps. For their standard forms, see figure <ref>.

Minimal surfaces in the three-dimensional Euclidean space 𝔼^3 are also classical  and important objects of study in differential geometry. On such minimal surfaces, isolated singular points, called branch points, where the tangent plane degenerates to a single point, often appear. 
To give some well-known historical examples of such surfaces, Catalan's minimal surface <cit.> and Henneberg's non-orientable minimal surface <cit.> are minimal surfaces with branch points locally having the diffeomorphism type f_D_4^-(u,v)=(uv,u^2- 3v^2,u^2v- v^3) called the D_4^- singularity, see Figure <ref> (and also Figure <ref> for D_4^- singularity). 






In this paper, as Lorentzian counter parts of the above surfaces, we deal with timelike minimal surfaces, which are surfaces in 𝕃^3 whose induced metric is Lorentzian and whose mean curvature vanishes identically. The aim of this paper is to show that when a surface is equipped with Lorentzian metric, the singularities appearing on it are quite different from those appearing on the above Riemmannian counter parts, namely minimal surfaces in 𝔼^3 and maximal surfaces in 𝕃^3.

As conformal minimal surfaces are characterized by the property that their coordinate functions are harmonic (i.e. the surface f satisfies the Laplace equation Δ f=0), each conformal timelike minimal surface f is also characterized by the wave equation f =0, where □ is the d'Alembert operator. For local null coordinates (u,v) so that f^*⟨, ⟩ =2Fdudv, the wave equation above is equivalent to f_uv=0. Here, ⟨,⟩ is the canonical Lorentzian inner product of 𝕃^3. Hence, arbitrary timelike minimal surface f is locally decomposed into the sum of two null curves as

    f(u,v) = φ(u)+ψ(v)/2,  ⟨φ', φ'⟩ = ⟨ψ', ψ'⟩ =0,
 
where φ'=dφ/du and ψ'=dψ/dv. For example, if we take two regular null curves φ(u)=(-u-u^3/3, u-u^3/3,u^2) and ψ(v)=(v+v^3/3, v-v^3/3, v^2), we obtain the timelike Enneper surface f=(φ(u)+ψ(v))/2. Singular points of this surface correspond to points (u,v) satisfying uv=-1 at which non-zero velocity vectors φ' and ψ' are linearly dependent. Singular points of this surface consist of cuspidal edges and swallowtails, see <cit.> for more details. 
As in this example, a timelike minimal surface with singularities that can be locally decomposed into a sum of regular null curves is called a minface, and singularities on minfaces are investigated by takahashi <cit.> (see also <cit.>) in the same way as maxfaces. 
In particular, the criteria for cuspidal edge, swallowtail and cuspidal cross cap are given in <cit.>. These are generic singularities on maximal surfaces, but if we consider more complicated diffeomorphism types, we will show in Section <ref> that singularities that do not appear on the maximal surfaces do appear on minfaces.

On the other hand, once we consider the situation where the generating null curves φ and ψ are no longer regular, more types of singularities start to appear on timelike minimal surfaces.
For example, let us consider the periodic closed null curve 

    γ(s)=∫(1, cos3s, sin3s)cos2sds

and the surface f(u,v)=(γ(u)+γ(v))/2. Since f is periodic in both u and v directions, f gives a timelike minimal torus, see Figure <ref>. The tangent plane of the surface f becomes zero-dimensional, i.e. df_p=0 at a point p where the null curves φ and ψ simultaneously fail to be regular, and f is locally diffeomorphic to the standard D_4^+ singularity f_D_4^+(u,v)=(uv,u^2+ 3v^2,u^2v+ v^3) at p. As shown in <cit.>, it is known that such D_4^+ singularities never appear on minimal or maximal surfaces. It has also been pointed out in <cit.> that timelike minimal surfaces with singularity where at least one of the generating null curves is not regular or smooth appear in string theory in physics.



In this paper, as seen in the example above, we consider a wider class of timelike minimal surfaces than minfaces and show that many diffeomorphism types of singularities appear on them that do not appear in the Riemannian counter parts. 

The main results of this paper are as follows.



 
  

  * Criteria for cuspidal butterfly and cuspidal S_1^+ singularity on timelike minimal surfaces are given, whereas the nonexistence of cuspidal S_1^- singularity, which usually appears on maximal surfaces, is shown on timelike minimal surfaces (Theorem <ref>).




  * A duality between cuspidal butterfly and cuspidal S_1^+ singularity holds: a timelike minimal surface f=(φ+ψ)/2 has a cuspidal butterfly at a singular point p if and only if its conjugate surface f̂:=(φ-ψ)/2 has a cuspidal S_1^+ singularity at p (Corollary <ref>). 




  * Criterion for (2,5)-cuspidal edge on timelike minimal surfaces is given and the existence of (2,2k+1)-cuspidal edge for arbitrary positive integer k is also shown (Theorem <ref> and Proposition <ref>). Furthermore, invariance of (2,5)-cuspidal edges under isometric and anti-isometric deformations is given (Corollary <ref>). 




  * Degenerate singularities (see Section <ref> for definition) are considered. In particular, criteria for cuspidal beaks and D_4^+ singularity are given, whereas the nonexistence of cuspidal lips and D_4^- singularity is shown on timelike minimal surfaces (Proposition <ref> and Proposition <ref>). Furthermore, invariance of D_4^+ singularities under isometric and anti-isometric deformations is given (Corollary <ref>). 




Finally, it is worth mentioning some related studies. While timelike minimal surfaces are locally represented as translation surfaces of null curves, singularities of translation surfaces in 𝔼^3 are studied in <cit.>. Singularities of timelike surfaces in 𝕃^3 with non-zero constant mean curvature are also studied in <cit.>. 





§ TIMELIKE MINIMAL SURFACES WITH SINGULARITIES



 §.§ Singularities on fronts and frontals

First we recall the singularity theory of frontals and fronts, see <cit.> for details. Let U be a domain in ℝ^2 and u, v are local coordinates on U. A smooth map f U ⟶ℝ^3 is called a frontal if there exists a smooth map n from U to the unit sphere S^2 such that n is perpendicular to df(TU) with respect to the Euclidean metric ⟨  , ⟩_E of ℝ^3. We call n the unit normal vector field of f. Moreover if the Legendrian lift L=(f,n) U ⟶ℝ^3× S^2 is an immersion, f is called a front. A point p∈ U where f is not an immersion is called a singular point of the frontal f, and we call the set of singular points of f the singular set. A singular point p is said to be of rank one (resp. rank zero) if rank(Jf_p)=1 (resp.  rank(Jf_p)=0). 




The function λ=(f_u, f_v, n) on (U;u,v) is called the signed area density function of the frontal f. A singular point p is called non-degenerate (resp. degenerate) if dλ_p ≠0 (resp. dλ_p=0). 

The set of singular points of f corresponds to zeros of λ. When a singular point p of a frontal f is non-degenerate, there exist an ε>0 and a regular curve γ=γ(t): (-ε, ε)⟶ U such that γ(0)=p and the image of γ coincides with the singular set of f around p. We call γ the singular curve and the direction of γ'=dγ/dt the singular direction. On the other hand, there exists a non-zero vector η∈Ker(df_p) because p is non-degenerate. We call η the null direction. 
 
The following functions on the singular curve γ are used in the criteria for various singularities.

    δ(t)=(γ'(t),η(t)),  ψ(t)=(df(γ'(t)),n(γ(t)),dn(η(t))).


Let U_i be domains of ℝ^2 and p_i be points in U_i (i=1, 2). Two smooth map germs f_1 (U_1,p_1)⟶ (ℝ^3,f_1(p_1)) and f_2  (U_2,p_2)⟶ (ℝ^3,f_2(p_2)) are 𝒜-equivalent if there exist diffeomorphism germs Φ (ℝ^2,p_1) ⟶ (ℝ^2,p_2) and Ψ (ℝ^3,f_1(p_1)) ⟶ (ℝ^3,f_2(p_2)) such that f_2 = Ψ∘ f_1∘Φ^-1. 

We set f^(2,2k+1)_CE, f_SW, f_CCR, f_CBF, f_CS_1^±, f_CBK, f_CL, f_D_4^± (ℝ^2,0)⟶ (ℝ^3,0) as

    f^(2,2k+1)_CE(u,v)=(u,v^2,v^2k+1),   f_SW(u,v)=(3u^4+u^2v,4u^3+2uv,v), 
       f_CCR(u,v)=(u,v^2,uv^3),   f_CBF(u,v)=(u,5v^4+2uv,4v^5+uv^2-u^2),  
       f_CS_1^±(u,v)=(u,v^2,v^3(u^2± v^2)),   f_CBK(u,v)=(v,u^3-uv^2,3u^4-2u^2v^2),
       f_CLP(u,v)=(v,u^3+uv^2,3u^4+2u^2v^2),   f_D_4^±(u,v)=(uv,u^2± 3v^2,u^2v± v^3),

where k is a positive integer. The map germs f^(2,2k+1)_CE, f_SW, f_CCR, f_CBF, f_CS_1^±, f_CBK, f_CL, f_D_4^± are called the (2,2k+1)-cuspidal edge, swallowtail, cuspidal cross cap, cuspidal butterfly, cuspidal S_1^± singularity, cuspidal beaks, cuspidal lips, D_4^± singularity, respectively (See Figure <ref>).









 §.§ Timelike minimal surfaces and their singularities

In this subsection, we clarify the notion of timelike minimal surfaces with singularities used here. First, we recall regular timelike minimal surfaces. 

A smooth immersion f from a 2-dimensional smooth manifold Σ to Minkowski 3-space 𝕃^3=(ℝ^3,⟨,⟩=-dt^2+dx^2+dy^2) is said to be timelike if its induced metric I=f̂⟨, ⟩ is Lorentzian. It is well-known that near each point of Σ, there exists a coordinate system (U;u,v), called a null coordinate system such that I=2Fdudv, where F is a smooth function. On U, f is a timelike minimal surface, that is, whose mean curvature vanishes identically if and only if it satisfies the linear wave equation f_uv=0. Hence, f can be locally decomposed into the sum of two regular null curves φ and ψ:

    f(u,v)=φ(u)+ψ(v)/2,

where a null curve is a curve whose tangent vector field is lightlike. Conversely, if f is written as (<ref>) using two linearly independent regular null curves φ and ψ, it is also a timelike minimal surface (cf. <cit.>).

 As well as maximal surfaces in 𝕃^3, timelike minimal surfaces have singularities in general. To deal with such singularities, Kim, Koh, Shin and Yang <cit.>  and Takahasi <cit.> (see also <cit.>) enlarged the class of timelike minimal surface as follows.

A non-constant smooth map fΣ⟶𝕃^3 is called a generalized timelike minimal surface if  f is immersed on an open dense set of Σ and there exists a local coordinate system (U;u,v) near each point of Σ such that ⟨ f_u, f_u ⟩ = ⟨ f_v, f_v ⟩ =0, and f_uv= 0 on U.
Moreover, if both f_u and f_v never vanish, f is called a minface.


The notion of minfaces is a timelike analogue of maxfaces introduced in <cit.> as a notion of spacelike maximal surfaces with rank one singularities. 

By definition, a generalized timelike minimal surface f is also written as (<ref>) although two null curves in  (<ref>) may not be linearly independent or not be regular. 
Since the generating curves φ and ψ in (<ref>) are null, f can be locally written as follows:

    f(u,v)
         =1/2∫^u_u_0( -1-(g_1)^2,1-(g_1)^2, 2g_1 )ω_1
       +1/2∫^v_v_0( 1+(g_2)^2,1-(g_2)^2, -2g_2 )ω_2 
        +f(u_0,v_0),

where g_1(u) and g_2(v) are functions, and 1-forms ω_1(u)=ω̂_1(u)du and ω_2(v)=ω̂_2(v)dv. The functions g_1 and g_2 are ℝ∪{±∞}-valued. The quadruple (g_1, g_2,ω_1, ω_2) is called the real Weierstrass data (or W-data for short) of f.
By (<ref>), W-data satisfies that if g_i(p)=±∞ for some i=1,2 then ω̂_̂î(p)=g_iω̂_̂î(p)=0 and g_i^2ω̂_i(p) is a real number.  

To avoid unnecessary complications, throughout this paper we always assume that 


 g_1 and g_2 have finite values at any singular point. 


This assumption holds automatically for any minface after taking an ambient isometry (see <cit.>).

The set of singular points, denoted by Σ(f), of a generalized timelike minimal surface f in (<ref>) is written as Σ(f)=Σ^g(f)∪Σ^ω(f), where

    Σ^g(f)={p∈Σ| g_1g_2(p)=1},  Σ^ω(f)={p∈Σ|ω̂_1ω̂_2(p)=0}.

We call a singular point in Σ^g(f) (resp.  in Σ^ω(f)) a g-singular point (resp. ω-singular point). 
The velocity vector fields of the generating null curves φ and ψ in (<ref>) are linearly dependent on Σ^g(f), while at least one of φ and ψ is not regular on Σ^ω(f). Therefore, Σ^g(f)∩Σ^ω(f) ≠∅ in general, and it is important when considering the types of singularities whether they belong to Σ^g(f)∩Σ^ω(f) or not as we will see in Section <ref>. By Definition <ref>, a minface is a timelike minimal surface without ω-singular points. 

We should remark that any generalized timelike minimal surface f becomes a frontal with the unit normal vector field

    n=1|n|_E(g_1+g_2,-g_1+g_2,1+g_1g_2),   |n|_E=√((1-g_1g_2)^2+2(g_1+g_2)^2)≠0.




§ SINGULARITIES IN Σ^G(F)∖Σ^Ω(F)

In this section, we consider timelike minimal surfaces with singularities in Σ^g(f)∖Σ^ω(f), such surfaces are nothing but minfaces in Definition <ref>. First, we recall criteria for cuspidal edge, swallowtail and cuspidal cross cap on minfaces given by Takahashi <cit.>. 


[<cit.>, see also <cit.>]
Let f U ⟶𝕃^3 be a minface with W-data (g_1, g_2,ω̂_1du, ω̂_2dv) on U, and p∈ U be a singular point. Then f is 𝒜-equivalent to 


  (i) the cuspidal edge at p if and only if (g_1)_u/g_1^2ω̂_1-(g_2)_v/g_2^2ω̂_2≠ 0 and (g_1)_u/g_1^2ω̂_1+(g_2)_v/g_2^2ω̂_2≠0 at  p,


  (ii) the swallowtail at p if and only if 
    (g_1)_u/g_1^2ω̂_1-(g_2)_v/g_2^2ω̂_2≠ 0, (g_1)_u/g_1^2ω̂_1+(g_2)_v/g_2^2ω̂_2=0, and g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u-g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v≠ 0 at p,



  (iii) the cuspidal cross cap at p if and only if 
    (g_1)_u/g_1^2ω̂_1-(g_2)_v/g_2^2ω̂_2=0, (g_1)_u/g_1^2ω̂_1+(g_2)_v/g_2^2ω̂_2≠0, and g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u+g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v≠0 at p.





Here, we use slightly different but equivalent criteria to those in <cit.>.


As shown in <cit.>, cuspidal edges, swallowtails and cuspidal cross caps are generic singularities on maximal surfaces and Fact <ref> shows that they also appear naturally on timelike minimal surfaces. However, as shown below, the appearance of other singularities is somewhat different from that of maximal surfaces. 





 §.§ Cuspidal butterfly and cuspidal S_1^± singularity

In this subsection, we give criteria for cuspidal butterfly and cuspidal S_1^+ singularity which appear in more complicated situations than those of the above singularities. Criteria for cuspidal buttefly and cuspidal S_1^± singularity on frontals were given by Izumiya and Saji <cit.> and Saji <cit.> as follows.

[<cit.>]
Let f U⊂ℝ^2 ⟶ℝ^3 be a front with the unit normal n and p∈ U be a non-degenerate singular point of f. Take a singular curve γ(t) with γ(0)=p and a null vector field η(t).
 Then, f is 𝒜-equivalent to the cuspidal butterfly at p if and only if δ(0)=δ'(0)=0 and δ”(0)≠0, where δ is the function defined in (<ref>).


[<cit.>]
Let f U⊂ℝ^2 ⟶ℝ^3 be a frontal with the unit normal n and p∈ U be a non-degenerate singular point of f. Take a singular curve γ(t) with γ(0)=p and a null vector field η(t).
 Then, f is 𝒜-equivalent to the cuspidal S_1^+ singularity (resp. the cuspidal S_1^- singularity) at p if and only if the followoing conditions hold.
 
 
  (i) δ(0)≠ 0,
 
  (ii) ψ(0)=ψ'(0)=0 and A:=ψ”(0)≠ 0,
 
  (iii) there exists a regular curve c (-ε,ε)⟶ U and l∈ℝ such that c(0)=p, c'(0) is parallel to η(0), ĉ”(0)≠ 0, ĉ”'(0)=ℓĉ”(0) and
 
    B:=(df(γ'(0)), ĉ”(0), 3ĉ^(5)-10ℓĉ^(4))≠0
 hold, where ĉ:=f∘ c,
 
  (iv) the product AB is positive (resp. negative). 
 


To use Fact <ref> for timelike minimal surfaces, we use the following Lemma.


Let fΣ→ℝ^3 be a frontal and p∈Σ be a non-degenerate singular point. Let γ=γ(t) be a singular curve with γ(0)=p and ξ, η be smooth vector fields near p so that ξ=γ', df(η)=0 and (ξ,η)>0 along γ. Assume that f satisfies the following conditions


  * η^3f(p)=0, 

  * ξ f(p) and η^2f(p) are linearly independent, and

  * ξη^3f(p) and η^2f(p) are linearly dependent.

Then there exists a regular curve c=c(t) satisfying the condition (iii) of Fact <ref> with B=3(ξ f, η^2f, η^5f)(p), where η^k f is the k-times derivative η⋯η f.


The proof of Lemma <ref> will be given in Appendix <ref>.


The proof is similar to the case of maximal surfaces in <cit.>. However, the condition ⟨ξ f(p), η^2f(p) ⟩_E =0 in <cit.> does not hold in general for the case of timelike minimal surfaces. So we need other assumptions as in Lemma <ref> showing the same conclusion.


By using the criteria above, we prove the following characterizations of the cuspidal butterfly and the cuspidal S_1^+ singularity on timelike minimal surfaces.


Let f and p be as in Fact <ref>.  Then f is 𝒜-equivalent to 


  (i) the cuspidal butterfly at p if and only if 

    (g_1)_u/g_1^2ω̂_1-(g_2)_v/g_2^2ω̂_2   ≠ 0,  (g_1)_u/g_1^2ω̂_1+(g_2)_v/g_2^2ω̂_2=0,  g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u-g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v= 0, and
       g_1/(g_1)_u[g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u]_u + g_2/(g_2)_v[g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v]_v≠ 0
     at p,



  (ii) the cuspidal S_1^+ singularity at p if and only if 

    (g_1)_u/g_1^2ω̂_1-(g_2)_v/g_2^2ω̂_2   =0,  (g_1)_u/g_1^2ω̂_1+(g_2)_v/g_2^2ω̂_2≠0,  g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u+g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v= 0, and
       g_1/(g_1)_u[g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u]_u - g_2/(g_2)_v[g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v]_v≠ 0
     at p,


Moreover, there is no cuspidal S_1^- singularity on any minface.



First, we prove the claim (i). Since p is a non-degenerate g-singular point, we can take a singular curve γ=γ(t) satisfying γ(0)=p and g_1(γ(t))g_2(γ(t))=1. Hence we can parametrize the singular curve γ and take a null vector field η as follows. 

    γ'=(g_2)_v/g_2∂_u-(g_1)_u/g_1∂_v,
      η=1/g_1ω̂_1∂_u+1/g_2ω̂_2∂_v
      on γ.

The functions δ and ψ defined in (<ref>) are written as

    δ=(g_1)_u/g_1^2ω̂_1+(g_2)_v/g_2^2ω̂_2,
      ψ=α((g_1)_u/g_1^2ω̂_1-(g_2)_v/g_2^2ω̂_2)
     on γ,

where α=-ω̂_1ω̂_2δ/2. For the function

    D:=g_1g_2/(g_1)_u(g_2)_vδ'
    =g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u-g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v,

δ'(0)=0 is equivalent to D(0)=0 and under this condition δ”(0)≠0 is also equivalent to D'(0)≠0.
By (i) in Fact <ref> and considering the quantity

    g_1g_2/(g_1)_u(g_2)_vD'=g_1/(g_1)_u[g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u]_u + g_2/(g_2)_v[g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v]_v,

we obtain the assertion (i). 

Next, we prove the assertion (ii).  Let us define the functions φ_i, ϕ_i and Φ_i (i=1,2) as follow.

    φ_i:=g_i'/g_i^2ω̂_i,  ϕ_i:=g_i/g_i'φ_i',  Φ_i:=g_i/g_i'ϕ_i',   i=1,2,

where the prime means the derivative with respect to u for i=1 and with respect to v for i=2, respectively.
By (<ref>), δ=φ_1+φ_2 and ψ=-1/2ω̂_1ω̂_2(φ_1+φ_2)(φ_1-φ_2) hold, and hence the conditions (i) and (ii) in Fact <ref> can be rewritten as

    φ_1=φ_2 ≠ 0,   ϕ_1=-ϕ_2,    A=-1/2ω̂_1ω̂_2(g_1)_u^2(g_2)_u^2(φ_1+φ_2)(Φ_1-Φ_2)≠ 0.

This means that the conditions (i) and (ii) in Fact <ref> are equivalent to the desired condition (ii) in Theorem <ref>. Thus, we show that the remaining conditions (iii) and (iv) with AB>0 in Fact <ref> follow automatically from the conditions (i) and (ii). 

By using ξ=γ' and η which are defined to satisfy equation (<ref>) and the relation g_1g_2=1 along γ, the vector fields ξ f, η^kf (1≤ k ≤ 5) and ξη^3f at p can be computed as follows.

    ξ f   =-ω̂_1ω̂_2/2(φ_1+φ_2)(g_1+g_2,g_1-g_2,-2),  η f=0,
    η^2f    = 1/2(φ_1+φ_2)(-g_1+g_2,-g_1-g_2,0),  η^3f = 0,
    η^4f    = 1/2[φ_1(φ_1^2+ϕ_1^2+φ_1Φ_1) + φ_2(φ_2^2+ϕ_2^2+φ_2Φ_2)](-g_1+g_2, -g_1-g_2,0)
       -3/2(φ_1^2ϕ_1+φ_2^2ϕ_2)(g_1+g_2,g_1-g_2,0),
    η^5f    =  -1/2[φ_1^2(φ_1^2+7ϕ_1^2+4φ_1Φ_1) - φ_2^2(φ_2^2+7ϕ_2^2+4φ_2Φ_2)](g_1+g_2, g_1-g_2,0)+ αη^2f,
    ξη^3f    = -3/2(g_1)_u(g_2)_v(φ_1ϕ_1+φ_2ϕ_2)(g_1+g_2, g_1-g_2,0)+βη^2f

at p, where α and β are real constants. Moreover, by using (<ref>), ξ f, η^2f, η^5f and ξη^3f are written as

    ξ f   =-ω̂_1ω̂_2φ_1(g_1+g_2,g_1-g_2,-2),  η^2f = φ_1(-g_1+g_2,-g_1-g_2,0), 
    η^5f    =  -2φ_1^3(Φ_1-Φ_2)(g_1+g_2, g_1-g_2,0)+ αη^2f,  ξη^3f =βη^2f.

Therefore, f satisfies the assumptions in Lemma <ref> and hence B is computed as

    B=3(ξ f, η^2f, η^5f)(p)= 
    -48ω̂_1ω̂_2φ_1^5(Φ_1-Φ_2).

Finally, by (<ref>) and (<ref>), the quantity AB in (iv) in Fact <ref> satisfies 

    AB=48ω̂_1^2ω̂_2^2(g_1)_u^2(g_2)_u^2φ_1^6(Φ_1-Φ_2)^2>0,

which proves the desired result.


Let us see that certain singularities on timelike minimal surfaces are transferred to each other by the following anti-isometric transformation called the conjugation.


Let (g_1, g_2,ω̂_1du, ω̂_2dv) be W-data of a generalized timelike minimal surface f. 

  * The one-parameter family {f_θ}_θ∈ℝ of generalized timelike minimal surafces with W-data (g_1, g_2,e^θω̂_1du, e^-θω̂_2dv) is called the associated family of f. 

  * The generalized timelike minimal surface f̂ with W-data (g_1, g_2,ω̂_1du, -ω̂_2dv) is called the conjugate surface of f.


The surfaces f and f_θ are isometric each other. The surfaces f and f̂ are anti-isometric each other, that is, their first fundamental forms I_f and I_f̂ satisfy I_f=-I_f̂. Hence, 
unlike the cases of minimal surfaces in 𝔼^3 and maximal surfaces in 𝕃^3, the conjugate surface of a timelike minimal surface does not belong to the associated family. Moreover, it is shown in <cit.> that away from flat points the associated families {f_θ} of f and {f̂_θ} of f̂ are the only isometric deformation and anti-isometric deformation of f, respectively, up to rigid motions in 𝕃^3. 

By Definition <ref> and Theorem <ref>, we obtain the following duality of singularities.

Let f be a timelike minimal surface and p be a singular point in Σ^g(f)∖Σ^ω(f). Then, f is 𝒜-equivalent to the cuspidal butterfly at p if and only if f̂ is 𝒜-equivalent to the cuspidal S_1^+ singularity at p.



Ogata and Teramoto <cit.> proved the duality between cuspidal butterflies and cuspidal S_1^- singularities on maximal surfaces. They also proved that there is no cuspidal S_1^+ singularity on these surfaces. In contrast, Theorem <ref> and Corollary  <ref> show a different aspect: cuspidal S_1^+ singularities appear on timelime minimal surfaces instead of cuspidal S_1^- singularities.


Let us consider the timelike minimal surface f with W-data

    g_1(u)=-cosu/1+sinu, g_2(v)=sinhϕ+cosvcoshϕ/sinv-coshϕ-cosvsinhϕ, 
    ω̂_1(u)=-μ/2(1+sinu), ω̂_2(v)=sinv-coshϕ-cosvsinhϕ/2μ,

where ϕ=arcsinh(1/2) and μ=(1+√(5))/2=coshϕ+sinhϕ.
Here, we show that the surface f is 𝒜-equivalent to the cuspidal butterfly at (u,v)=0. Since g_1g_2=1 at 0 and the quantities in Theorem <ref> are computed as

    (g_1)_u/g_1^2ω̂_1+   (g_2)_v/g_2^2ω̂_2
    =
    g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u
    =
    g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v
    =0,  (g_1)_u/g_1^2ω̂_1+(g_2)_v/g_2^2ω̂_2=-4/μ≠0,   
        and g_1/(g_1)_u[g_1/(g_1)_u((g_1)_u/g_1^2ω̂_1)_u]_u + g_2/(g_2)_v[g_2/(g_2)_v((g_2)_v/g_2^2ω̂_2)_v]_v=2≠ 0
     at 0,

f is 𝒜-equivalent to the cuspidal butterfly at 0. By the duality in Corollary <ref>, the conjugate surface f̂ is 𝒜-equivalent to the cuspidal S_1^+ singularity at 0 (see Fig. <ref>).
We remark that the surface f belongs to the associated family of a timelike minimal surface with planar curvature lines studied in <cit.>. 










§ SINGULARITIES IN Σ^Ω(F)

In this section, we give some properties of ω-singular points on a generalized timelike minimal surface.

Let p∈Σ^ω(f) be a ω-singular point satisfying rank(df_p)=1 of a generalized timelike minimal surface f with W-data (g_1, g_2,ω̂_1du, ω̂_2dv). Then the following statements hold.


  (i) f is a front at  p with ω̂_1(p)=0 (resp. with ω̂_2(p)=0) if and only if (g_1)_u≠ 0 (resp. (g_2)_v≠ 0) at p.

  (ii) The singular point p with ω̂_1(p)=0 (resp. with ω̂_2(p)=0) is non-degenerate if and only if (ω̂_1)_u(p)≠ 0 (resp. (ω̂_2)_v(p)≠ 0) and g_1g_2(p)≠1.




By taking the derivatives of the equation (<ref>), we have 

    n_u=(n_u^1,n_u^2,n_u^3)=(g_1)_u/|n|_E^3((1+g_2^2)(1-g_1g_2),-1-3g_1g_2-3g_2^2-g_1g_2^3,2(g_2^3-g_1)), 
    
    n_v=(n_v^1,n_v^2,n_v^3)=(g_2)_v/|n|_E^3((1+g_1^2)(1-g_1g_2),1+3g_1g_2+3g_1^2+g_1^3g_2,2(g_1^3-g_2)) .

Since f_u(p)=0 for p∈Σ^ω(f) with ω̂_1(p)=0, f is a front at p if and only if n_u(p)≠0. By (<ref>), n_u(p)≠0 is equivalent to the condition (g_1)_u(p)=0. Then, we have proved (i) for the case ω̂_1(p)=0. Finally, the signed area density function λ can be written as

    λ=Λ(1-g_1g_2)ω̂_1ω̂_2,  Λ=-√((1-g_1g_2)^2+2(g_1+g_2)^2)/2.

Since ω̂_1(p)=0 we have dλ_p=Λ(p)(1-g_1g_2(p))(ω̂_1)_u(p)ω̂_2(p), and hence p is non-degenerate if and only if (ω̂_1)_u≠ 0 and g_1g_2≠1 at p. The proof of (i) and (ii) for the case ω̂_2(p)=0 is same.


In the following sections, when considering an ω-singular point p satisfying rank(df_p)=1, we will state the assertions only for the case ω̂_2(p)=0.




 §.§ (2,2k+1)-cuspidal edges

In this section, we consider (2,2k+1)-cuspidal edge of the form:

    (u,v)⟼ (u,v^2,v^2k+1),    k=1,2,⋯

In general, if we consider a cuspidal edge of the form f(u,v)=(u,v^m,v^n) (2≤ m≤ n) and assume that it is non-degenerate, then we have m=2. Hence, (<ref>) means that we consider arbitrary non-degenerate cuspidal edges.

Usually the (2,3)-cuspidal edge is simply called the cuspidal edge. First, we give a characterization of cuspidal edges as ω-singular points.


The cuspidal edge is only non-degenerate rank one ω-singular point on which the considered generalized timelike minimal surface is a front.



For an ω-singular point p=(0,0) with ω̂_2(p)=0 and rank(df_p)=1, we can take the singular curve γ(u)=(u,0) and the null direction given by η=∂/∂v, which satisfies (γ',η)≠0. By the useful criterion proved in <cit.>, we prove the desired result.


By Lemma <ref>, any rank one non-degenerate ω-singular point on which the considered surface is a front becomes a cuspidal edge automatically. On the other hand, it is natural to ask what is a typical rank one non-degenerate ω-singular point on which the considered surface is not a front. As an answer to this question, we prove the following theorem.


Let p∈Σ^ω(f) be a ω-singular point satisfying rank(df_p)=1 and ω̂_2(p)=0 of a generalized timelike minimal surface f=f(u,v) with W-data (g_1, g_2,ω̂_1du, ω̂_2dv). If f is not a front at p, then f is 𝒜-equivalent to the (2,5)-cuspidal edge if and only if ((g_2)_vv/(ω̂_2)_v)_v(p)≠0.



To prove Theorem <ref>, we use the following criterion for (2,5)-cuspidal edges.
[<cit.>]
Let U be a domain of ℝ^2,
f U →ℝ^3 a frontal,
and p ∈ U a non-degenerate singular point of the first kind.
Moreover, let
γ(t) (|t|<ε) 
be a singular curve passing through p=γ(0)
and η(t) a null vector field along γ.
Take smooth vector fields ξ=ξ(u,v) and η=η(u,v) on U
which are extensions of γ'(t) and η(t), respectively.
Then, p=γ(0) is a (2,5)-cuspidal edge 
if and only if 

    (ξ f, η^2f, η^3f)=0
       on γ,    and  
      
    (ξ f, η̃^2 f,  3η̃^5 f - 10 C η̃^4 f)(p)≠0

hold, where η^k f implies k-times derivative η⋯η f, 
η̃ is a special null vector field satisfying

    ⟨ξ f, η̃^2 f⟩_E(p)=
      ⟨ξ f, η̃^3 f⟩_E(p)=0

(⟨, ⟩_E means the Euclidean inner product),

and C is a constant such that

    η̃^3 f(p) = C η̃^2 f(p).




Now we can take the vector field η̃ and the constant C as follows.

Let f and p be the same as in Theorem <ref>, and assume that p=0. Then we can take the vector filed η̃ and the constant C in Fact <ref> as follows.

    η̃(u,v)=(av+bv^2) ∂/∂ u+∂/∂ v   and    C=(ω̂_2)_vv/(ω̂_2)_v(p),

where the constants a and b are

    a=(ω̂_2)_v/ω̂_1(g_1+g_2/1+g_1^2)^2(p)    and    b=(ω̂_2)_vv/2ω̂_1(g_1+g_2/1+g_1^2)^2(p).




As proved in <cit.>, we can construct η̃ satisfying the condition (<ref>) as 

    η̃(u,v)=(av+bv^2) ∂/∂ u+∂/∂ v, where a=-⟨ f_u,f_vv⟩_E/⟨ f_u,f_u⟩_E(p) and  b=-⟨ f_u,f_vvv⟩_E/2⟨ f_u,f_u⟩_E(p).


By the assumption and Lemma <ref>, the relations ω̂_2(p)=g_2'(p)=0 hold and hence we can calculate 

    f_u=ω̂_1/2   (-1-g_1^2, 1-g_1^2, 2g_1),
    
    f_vv=(ω̂_2)_v/2(1+g_2^2,1-g_2^2, -2g_2),      f_vvv=(ω̂_2)_vv/2(1+g_2^2,1-g_2^2, -2g_2)  at p.

By using them, 

    ⟨ f_u,f_u⟩_E   =ω̂_1/2(1+g_1^2)^2,
    ⟨ f_u,f_vv⟩_E=-ω̂_1(ω̂_2)_v/2(g_1+g_2)^2,     ⟨ f_u,f_vvv⟩_E=-ω̂_1(ω̂_2)_vv/2(g_1+g_2)^2

hold at p. Thus, the constants a and b are written as (<ref>).

For η̃=(av+bv^2) ∂/∂ u+∂/∂ v, η̃^2f(p) and η̃^3f(p) are written as

    η̃^2f=af_u+f_vv,  η̃^3f=2bf_u+f_vvv  at p.

Based on the relations 

    f_vvv(p)=(ω̂_2)_vv/(ω̂_2)_vf_vv(p)  and   2b=(ω̂_2)_vv/(ω̂_2)_va,

we obtain the desired constant C.


By using Fact <ref> and Lemma <ref>, we can prove Theorem <ref> as follows.

Without loss of generality, we may assume that p=(0,0).
First we check the condition (<ref>). Since the singular curve passing through p is γ(u)=(u,0), we can take the vector fields ξ and η as ξ=∂/∂ u and η=∂/∂ v.
As we saw in the proof of Lemma <ref>, η^2f=f_vv and η^3f=f_vvv are parallel on γ. Therefore we have checked the condition (<ref>). 

Next, we see the quantity (<ref>).
By using Lemma <ref>, we can calculate that

    η̃^4f   =3a^2f_uu+f_vvvv
       =3a^2[(ω̂_1)_u/2c_1-ω̂_1(g_1)_uc_3]
    +(ω̂_2)_vvv/2c_2-3(ω̂_2)_v(g_2)_vvc_4,
    η̃^5f=   20abf_u+f_vvvvv
    
    =   20ab[(ω̂_1)_u/2c_1-ω̂_1(g_1)_uc_3]+(ω̂_2)_vvvv/2c_2-[6(ω̂_2)_vv(g_2)_vv+4(ω̂_2)_v(g_2)_vvv]c_4

at p, where

    c_1=(-1-g_1^2(p),1-g_1^2(p),2g_1(p)),  c_2=(1+g_2^2(p),1-g_2^2(p),-2g_2(p)), 
    c_3=(g_1(p),g_1(p),-1),  c_4=(-g_2(p),g_2(p),1).

Nothing that ξ and η̃^2f at p were calculated in the proof of Lemma <ref> as follows

    ξ f=ω̂_1/2c_1   and   η̃^2f=a(ω̂_1)/2c_1+(ω̂_2)_v/2c_2,

we obtain the following equations at p

    (ξf,η̃^2f,-10Cη̃^4f)=-5Cω̂_1(ω̂_2)_v/2(c_1,c_2,η̃^4f)
       =5ω̂_1(ω̂_2)_vv/2[3a^2 ω̂_1(g_1)_u (c_1,c_2,c_3)+3(ω̂_2)_v(g_2)_vv(c_1,c_2,c_4)]
       =15a^2(ω̂_1)^2 (ω̂_2)_vv (g_1)_u (1-g_1g_2)^2 - 15ω̂_1(ω̂_2)_v (ω̂_2)_vv (g_2)_vv (1-g_1g_2)^2,

and 

    (ξf,η̃^2f,3η̃^5f)   =3ω̂_1(ω̂_2)_v/4(c_1,c_2,η̃^5f)   
       =-3ω̂_1(ω̂_2)_v/4[20ab ω̂_1(g_1)_u (c_1,c_2,c_3)   
        + ( 6(ω̂_2)_vv(g_2)_vv+4(ω̂_2)_v(g_2)_vvv)(c_1,c_2,c_4)]   
       =-15a^2(ω̂_1)^2 (ω̂_2)_vv (g_1)_u (1-g_1g_2)^2
          + ω̂_1(ω̂_2)_v ( 9(ω̂_2)_vv(g_2)_vv+6(ω̂_2)_v(g_2)_vvv)(1-g_1g_2)^2,

where we used the relations 2ab=a^2(ω̂_2)_vv/(ω̂_2)_v, see Lemma <ref>, and (c_1,c_2,c_3)=-(c_1,c_2,c_4)=2(1-g_1g_2)^2 in the last equality. Therefore, we have


    (ξ f, η̃^2f,  3η̃^5 f - 10 C η̃^4 f)=   6ω̂_1(ω̂_2)_v (1-g_1g_2)^2( (ω̂_2)_vv(g_2)_vvv-(ω̂_2)_vv(g_2)_vv)
    
     =   6ω̂_1(ω̂_2)_v^3 (1-g_1g_2)^2( (g_2)_vv/(ω̂_2)_v)_v at p.

Noting that ω̂_1(ω̂_2)_v^3 (1-g_1g_2)^2≠ 0 at p due to non-degeneracy of p and Lemma <ref>, we have the desired result.


As a corollary of Theorem <ref>, we obtain the following invariance property of (2,5)-cuspidal edges under the isometric and anti-deformtaions introduced in Definition <ref>.

Let p be an ω-singular point of a generalized timelike minimal surface f. Then, the following statements are equivalent.


  (i) f is 𝒜-equivalent to the (2,5)-cuspidal edge at p,

  (ii) f_θ is 𝒜-equivalent to the (2,5)-cuspidal edge at p for arbitrary θ∈ℝ and 

  (iii) f̂ is 𝒜-equivalent to the (2,5)-cuspidal edge at p.



This is an interesting phenomenon by the following reasons:
Honda, Koiso and Saji <cit.> pointed out that maxfaces cannot have (2,5)-cuspidal edges because if we assume that a singular point p on a maxface satisfies the condition of (2,5)-cuspidal edges, we can prove that p becomes a fold singularity. On the other hand, they also proved in <cit.> that spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces, for short) do not admit fold singularities and there is a duality of conlike singularities of Delaunay surfaces and (2,5)-cuspidal edges of their conjugate surfaces. However Theorem <ref> and Corollary <ref> show that for the timelike case, (2,5)-cuspidal edges appear frequently in the sense of Theorem <ref> even for zero mean curvature surfaces, and such singular points have a self duality which is different from that of spacelike CMC surfaces.
Moreover, in contrast to the case of maximal surfaces, the following proposition shows that timelike minimal surfaces can have any type of (2,2k+1)-cuspidal edges, that is, cuspidal edges can be as steep as desired.


For any positive integer k, the timelike minimal surface f with W-data (g_1, g_2,ω̂_1du, ω̂_2dv) satisfying

g_2(v)=v^2k-1, ω̂_1(0)≠0 and ω̂_2(v)=v,

has (2,2k+1)-cuspidal edges on v=0.



After a translation and the local diffeomorphism  
Φ_1(X,Y,Z)=(-X+Y,Y,Z),
the timelike minimal surface f defined by (<ref>) is written by

    f_1(u,v)= Φ_1^-1∘ f(u,v)=1/2∫^u_0( 2,1-(g_1)^2, 2g_1 )ω_1+( -v^4k/4k,v^2/4-v^4k/8k,v^2k+1/2k+1).

  We set (γ^0,γ^1,γ^2)=( -v^4k/4k,v^2/4-v^4k/8k,v^2k+1/2k+1) and consider the new parameter V=vσ(v), σ(v)=(1/4-v^4k-2/8k)^1/2 near v=0.
 Since 
 
    dγ^0/dV(V)=-v^4k-1τ(v),  τ(v)=dv/dV=(σ(v)+vσ'(v))^-1,

 we obtain the Taylor expansion γ^0(V)=∑^∞_i=4ka_iV^i, a_4k≠ 0.

 By separating the even and odd terms, we can take functions h_1 and h_2 satisfying h_1(0)=h_1'(0)=h_2(0)=0 such that γ^0(V)=h_1(V^2)+V^2k+1h_2(V^2). Similarly, γ^2 is also expressed as
 
    γ^2(V)=b_2k+1V^2k+1+b_2k+1V^2k+1h_3(V^2)+h_4(V^2),

 where b_2k+1≠ 0 and h_3, h_4 are functions satisfying h_3(0)=h_4(0)=0.
 
 Therefore, if we put the new parameter U=∫^u_0ω_1, the map f_1 is written as
 
    f_1(U,V)   =(U,f^1_1(U),f^2_1(U))
       +(h_1(V^2)+V^2k+1h_2(V^2), V^2, b_2k+1V^2k+1+ b_2k+1V^2k+1h_3(V^2)+h_4(V^4) ),

 where f^1_1 and f^2_1 are functions with f^1_1(0)=f^2_1(0)=0.
Since the map
 
    Φ_2(X,Y,Z)=(X+h_1(Y)+Zh_2(Y),f^1_1(X)+Y,f^2_1(X)+b_2k+1Zh_3(Y)+h_4(Y)),

 satisfies Φ_2(0)=0 and the Jacobian of Ψ_2 at 0 is b_2k+1-b_2k+1h_1'(0)(f^1_1)'(0)=b_2k+1, it is a local diffeomorphism of ℝ^3 at 0. Hence we conclude that 
 
    Φ_2^-1∘ f_1(U,V)=(U,V^2,V^2k+1).






 §.§ Cuspidal beaks and lips
 Curspidal beaks and cuspidal lips are degenerate rank one singular points of fronts,  and criteria for them were given by Izumiya, Saji and Takahasi <cit.> as follows.
[<cit.>]
Let f U⊂ℝ^2 ⟶ℝ^3 be a front and p∈ U be a singular point of f. The map f is 𝒜-equivalent to the cuspidal beaks (resp. cuspidal lips) at p if and only if rank(df_p)=1, dλ_p=0,  ηηλ(p)≠0 and (λ)(p)<0 and (resp. rank(df_p)=1, dλ_p=0 and (λ)(p)>0) hold, where (λ) means the determinant of the Hessian matrix of the function λ.

Applying Fact <ref>, we have the following. 

Let p ∈Σ^ω(f) be an ω-singular point satisfying ω̂_2(p)=0 of a generalized timelike minimal surface f with W-data (g_1, g_2,ω̂_1du, ω̂_2dv). Then f is 𝒜-equivalent to the cuspidal beaks if and only if the following conditions hold:

    ω̂_1≠0, (ω̂_2)_v≠ 0, g_1g_2=1  and  (g_1)_u(g_2)_v≠ 0  at p.

Moreover, there is no cuspidal lips on any generalized timelike minimal surface in 𝕃^3.


By Lemma <ref>, f is a front at p if and only if ω̂_1(p)≠ 0 and (g_2)_v(p)≠ 0. Since p is degenerate, (ω̂_2)_v(p) =0 or g_1(p)g_2(p)=1 also holds. Therefore, the function λ in (<ref>) satisfies λ_u=λ_v=λ_uu=0 at p. Hence, the non-existence of cuspidal lips is proved by Fact <ref> and the relation (λ)(p)=-λ_uv(p)^2≤ 0. Since 

    λ_uv=-Λ(g_1)_ug_2ω̂_1(ω̂_2)_v+Λ_u(1-g_1g_2)ω̂_1(ω̂_2)_v  at p,

it is sufficient to consider the case of (ω̂_2)_v(p)≠ 0 and g_1(p)g_2(p)=1.
In this case, λ_uv=-Λ(g_1)_ug_2ω̂_1(ω̂_2)_v at p and hence the condition (λ)(p) < 0 is attained only when (g_1)_u(p)≠0. Moreover, the null vector field η=∂_v satisfies

    ηηλ=λ_vv=-Λ g_1(g_2)_vω̂_1(ω̂_2)_v ≠ 0  at p,

which proves the desired result by Fact <ref>.




 §.§ D_4^± singularities

D_4 singularities are typical rank zero singularities and consequently they are degenerate. In fact, as shown in Introduction, they appear on minimal surfaces in 𝔼^3 as branch points.
First we give a necessary and sufficient condition for a generalized timelike minimal surface to be a front at a rank zero singular point. 

Let f=f(u,v) be a generalized timelike minimal surface with W-data (g_1, g_2,ω̂_1du, ω̂_2dv). The following statements hold.


  (i) A point p satisfies df_p=0 if and only if ω̂_1(p)=ω̂_2(p)=0.

  (ii) Let p be a singular point with df_p=0. Then, the surface f is a front at  p if and only if g_1g_2(p)≠ 1 and (g_1)_u(g_2)_v(p)≠ 0.



Since 

f_u=ω̂_1(-1-g_1^2, 1-g_1^2, 2g_1)/2 and f_v=ω̂_2(1+g_2^2, 1-g_2^2, -2g_2)/2,

a point p satisfies df_p=0 if and only if ω̂_i(p)=g_iω̂_̂î(p)=g_i^2ω̂_i(p)=0 for each i=1,2. It is equivalent to ω̂_i(p)=0 by the assumption that g_i(p)∈ℝ. Then we have proved the assertion (i). 

Next we prove the assertion (ii). Since df_p=  0, f is a front at p if and only if n_u(p) and n_v(p) are linearly independent. By (<ref>) and (<ref>), we have

    n_u× n_v=(g_1)_u(g_2)_v(1-g_1g_2)ñ,  ñ is a non-zero vector,

which proves the assertion.


Saji <cit.> gave useful criteria for D_4 singularities on fronts.
[<cit.>]
Let f U⊂ℝ^2 ⟶ℝ^3 be a front and p be a point in U. Then f is 𝒜-equivalent to the D_4^+ singularity (resp. D_4^- singularity) if and only if df_p=0 and (λ)(p)<0 (resp. (λ)(p)>0) hold,
where (λ) means the determinant of the Hessian matrix of the function λ.

Applying Fact <ref>, we have the following. 


Let f=f(u,v) be a generalized timelike minimal surface with W-data (g_1, g_2,ω̂_1du, ω̂_2dv). Then f is 𝒜-equivalent to the D_4^+ singularity if and only if

    ω̂_1(p)=ω̂_2(p)=0, g_1g_2(p)≠1, (g_1)_u(g_2)_v(p)≠0   and  (ω̂_1)_u(ω̂_2)_v(p)≠ 0.

Moreover there is no D_4^- singularity on any generalized timelike minimal surface in 𝕃^3.



By Lemma <ref>, f is a front at p with df_p=0 if and only if 

    ω̂_1(p)=ω̂_2(p)=0, g_1g_2(p)≠1  and  (g_1)_u(g_2)_v(p)≠0.

The signed area density function λ in (<ref>) can be rewritten as

    λ=Λ̃ω̂_1ω̂_2,  Λ̃=-(1-g_1g_2)√((1-g_1g_2)^2+2(g_1+g_2)^2)/2,
 
where Λ̃ satisfis Λ̃(p)≠0 by the front condition (<ref>). Since

    λ_u=Λ̃_uω̂_1ω̂_2+Λ̃(ω̂_1)_uω̂_2,  λ_v=Λ̃_vω̂_1ω̂_2+Λ̃ω̂_1(ω̂_2)_v,

we have 

    λ_uu=λ_vv=0,  λ_uv=Λ̃(ω̂_1)_u(ω̂_2)_v  at p.

Hence,

    (λ)=-λ_uv^2=-Λ̃^2(ω̂_1)_u^2(ω̂_2)_v^2  at p.


By (<ref>), (<ref>) and Fact <ref>, we obtain the desired criterion. Moreover the right hand side of the equation (<ref>) is always nonpositive. This means that there is no D_4^- singularity on any generalized timelike minimal surface by Fact <ref>.
 


The Hopf differentials of f are defined by 

    Qdu^2=⟨ f_uu, n_L ⟩ du^2,   Rdv^2=⟨ f_vv, n_L ⟩ dv^2

where n_L is a unit normal vector field of f in 𝕃^3. By a direct calculation, we can see that Q and R are written as Q=ω̂_1(g_1)_u and R=-ω̂_2(g_2)_v for an appropriate choice of n_L and these relations make it possible to define Q and R across singular points. 
By using these Q and R, the relation (<ref>) is equivalent to

    df_p=0, p ∈Σ^ω(f)∖Σ^g(f)  and  Q'R'(p)≠0.




Ogata and Teramoto <cit.> gave a criterion for D_4^- singularities on surfaces  (resp. spacelike surfaces) in 3-dimensional Riemannian (resp. Lorentzian) spaceforms with constant mean curvature. They also show that there is no D_4^+ singularity on these surfaces, but the opposite phenomenon occurs for timelike surfaces in the sense of Proposition <ref>.


By proposition <ref>, we also have the following invariance of D_4^+ singularities as well as (2,5)-cuspidal edges in Corollary <ref>.

Let p be an ω-singular point of a generalized timelike minimal surface f. Then, the following statements are equivalent.


  (i) f is 𝒜-equivalent to the D_4^+ singularity at p,

  (ii) f_θ is 𝒜-equivalent to the D_4^+ singularity at p for arbitrary θ∈ℝ and 

  (iii) f̂ is 𝒜-equivalent to the D_4^+ singularity at p.



We remark that the same invariance of D_4^- singularities is also valid for generalized maximal surfaces in 𝕃^3.
The following torus was found by Kim, Koh, Shin and Yang <cit.>.

Let us consider the 2π-periodic null curve 

    γ(t)=(1/2sin2t,1/2sint+1/6sin3t,1/2cost-1/6cos3t),

and the generalized timelike minimal torus f(u,v)=(γ(u)+γ(v))/2, (u,v)∈ [0,2π]^2.
This torus is found by Kim Koh Shin and Yang <cit.> and we will show this surface has D_4^+ singularities.
The singular points at which df=0 are

    (u,v)=((1+2k)π/4,(1+2l)π/4),   0≤ k,l ≤ 3.

The W-data (g_1, g_2,ω̂_1du, ω̂_2dv) of f is determined as

    g_1(u)=sinu/cosu-1   , g_2(v)=-sinv/cosv+1,
    ω̂_1(u)=cos2u(cosu-1)/2   , ω̂_2(v)=cos2v(cosv+1)/2.

Here, we remark that the functions g_1 and g_2 have finite values at above singular points. We can see that W-data satisfies (g_1)_u(g_2)_v(ω̂_1)_u(ω̂_2)_v≠0 at singular points (<ref>). Since g_1((1+2k)π/4)g_2((1+2l)π/4)≠ 1 for k≠ l, Proposition <ref> shows that f has these twelve D_4^+ singularities (see Figure <ref>).
 
Next we consider its conjugate surface. Since the generalized timelike minimal torus f has the fold singular points (see <cit.> for details) 

    Γ={(u,u)| u ∈ [0,2π], u≠π/4,3π/4,5π/4,7π/4}

with the folded symmetry f(v,u)=f(u,v), its conjugate surface f̂ has conelike singular points satisfying f̂(Γ)=0 in 𝕃^3 with the point symmetry f̂(v,u)=-f̂(u,v). By Corollary <ref>, the D_4^+ singularities on f correspond to those of f̂, see Figure <ref>. 












§ PROOF OF LEMMA <REF>

In this section, we give a proof of Lemma <ref>. Although the strategy of the proof is the same as in the case of maximal surfaces <cit.>, it should be noted that different assumptions are adopted, see also Remark <ref>.

We can take a local coordinates (U; σ,τ) around p such that ξ = ∂_σ and η=∂_τ. 
Let us define the function F(σ,τ):=⟨ f(σ,τ), f_σ(0)⟩_E. Since F_σ(0)=⟨ f_σ(0), f_σ(0)⟩_E≠ 0, there exists a function σ=σ(τ) such that F(σ(τ),τ)=0 and σ(0)=0. We show the curve c(τ)=(σ(τ),τ) satisfies the condition (iii) of Fact <ref>.

Taking the derivative of F∘ c with respect to τ, we have 

    0=d(F∘ c)/dτ = ⟨ f_σ(σ,τ)σ'(τ)+f_τ(σ,τ), f_σ(0)⟩_E,

where σ'=dσ/dτ. Since f_τ=η f =0 along σ-axis, we have σ'(0)=0 and hence c'(0)=η(0). 
Based on the relations σ'(0)=0 and f_στ =0 along σ-axis, we can compute 

    0=d^3(F∘ c)/dτ^3 (0)= ⟨ f_σ(0)σ”'(0)+f_τττ(0), f_σ(0)⟩_E.

By the assumption  f_τττ(0)=η^3f(p)=0, we obtain the third derivative relation σ”'(0)=0. 
We note that, unlike the situation in <cit.>, σ”(0)=0 does not follow under the current assumption.

Next we check the condition (iii) of Fact <ref> for ĉ:=f∘ c. Taking the derivatives of ĉ, we have

    ĉ'   =f_σσ'+f_τ,
    ĉ”   =f_σσ(σ')^2+2f_στσ'+f_σσ”+f_ττ,
    ĉ”'   =f_σσσ(σ')^3+3f_σστ(σ')^2+3f_σττσ'+3f_σσσ'σ”+3f_στσ”+f_σσ”'+f_τττ.

So, by the relations σ'(0)=σ”'(0)=0 and f_στ(0)=f_τττ(0)=η^3f(p)=0, we obtain ĉ”'(0)=0 and ĉ”(0)=f_σ(0)σ”(0)+f_ττ(0). By the assumption that ξ f(p) and η^2f(p) are linearly independent, ĉ”(0)≠0. Thus, we can take the constant ℓ in (iii) of Fact <ref> as ℓ=0. Moreover, the fifth derivative c^(5)(0) is also computed as

    c^(5)(0)=10ξη^3 f(p)+σ^(5)(0)ξ f(p)+ η^5f(p).

Finally, we obtain

    B=(ξ f(p), ĉ”(0), 3ĉ^(5)-10ℓĉ^(4))=3(ξ f(p),η^2f(p), η^5f(p)),

where we used the assumption that ξη^3f(p) and η^2f(p) are linearly dependent.



Acknowledgement
This work was partially supported by JSPS KAKENHI Grant Numbers 19K14527 and 23K12979. The author express his gratitude to Professors Atsufumi Honda, Miyuki Koiso, Yuta Ogata, Kentaro Saji, Masatomo Takahashi and Keisuke Teramoto for their valuable comments and suggestions. In particular, he would also like to express his sincere thanks to Professors Kentaro Saji and Masatomo Takahashi again for the opportunity to speak on topics related to this study at Geometry Seminar at Kobe University and series of lectures at Muroran University, and to the participants who attended those seminars.

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 Paper Title: Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms.










2000 Mathematics Subject Classification: 53C40,  53C42.
Key words and phrases: quaternionic space form, Laplacian,
curvature-adapted hypersurface, finite Chen-type, tube about a submanifold, principal curvatures of a hypersurface.


§ ABSTRACT 
In a curvature-adapted hypersurface M of a quaternionic-Kähler manifold M  
the maximal quaternionic subbundle 𝒟 of TM and its orthogonal complement 𝒟^⊥ in TM are invariant subspaces of the shape operator at each point.
We classify curvature-adapted real hypersurfaces M of non-flat quaternionic space forms ℍP^m and ℍH^m that are 
of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where in the hyperbolic case we assume additionally that the hypersurace has constant principal curvatures. The 
position vector of a such submanifold in the ambient (pseudo) Euclidean space is decomposable into a sum of a constant vector and two nonconstant 
vector eigenfunctions of the Laplace operator of the submanifold  belonging to different eigenspaces. In the quaternionic projective space they include geodesic hyperspheres of arbitrary radius r ∈ (0, π/2)
except one, two series of tubes about canonically embedded quaternionic projective spaces of lower dimensions and two particular tubes about a
canonically embedded ℂP^m ⊂ℍP^m.
 On the other hand, the list of 2-type 
curvature-adapted hypersurfaces with constant principal curvatures in ℍH^m is reduced to geodesic spheres and tubes of arbitrary radius about totally geodesic 
quaternionic hyperplane ℍH^m-1. Among these hypersurfaces we determine those that are mass-symmetric or minimal. We also show that the horosphere H_3 in ℍH^m is not of finite type but satisfies Δ^2x = const.



Curvature-Adapted Hypersurfaces of 2-type

in non-flat quaternionic Space Forms
    Ivko Dimitrić
    August 5, 2024
===============================================================================






§ INTRODUCTION


0.4truecm
In the theory of immersions of finite type developed by B. Y. Chen <cit.>, 
a submanifold M^n of Euclidean or pseudo-Euclidean space isometrically immersed by x: M^n → E^N_(K) is said to be of finite type
in E^N_(K) if the position vector x can be decomposed into a finite sum of vector eigenfunctions of the Laplacian Δ _M
on M, viz.

    x = x_0 + x_t_1 + … + x_t_k;       x_t_i = λ_t_i x_t_i,    i= 1,...,k,   λ_t_i = const ,

where x_0 = const,   x_t_i≠const, and the Laplacian acts on a vector-valued function 
componentwise. If λ_t_i∈ R are all different, the immersion is said to be of Chen-type k  or simply of k-type. This 
definition is a generalization of the notion of minimal submanifolds of a  hypersphere or the ambient Euclidean space, whose
Chen-type is 1. For a compact submanifold, the constant part x_0 is the center of mass and if x immerses M^n into a central hyperquadric
of a Euclidean or pseudo-Euclidean space, the immersion is said to be mass-symmetric in that hyperquadric if x_0 coincides with the center of the said hyperquadric. Moreover, decomposition (<ref>) also makes sense for noncompact submanifolds, but x_0 may not be uniquely 
determined, namely in the case when one of the eigenvalues λ_t_i above is zero (This is the case of submanifolds of null k-type).

This notion can be extended to submanifolds
x: M^n →M of a more general manifold M as long as there is a
“reasonably nice" (typically equivariant, with parallel second fundamental form) embedding  Φ: M→ E^N_(K) of the ambient 
manifold M into a suitable (pseudo) Euclidean space, in which case M is said to be of Chen-type k (via Φ) if the 
composite immersion x:=Φ∘ x is of Chen-type k.
Symmetric spaces of rank 1 (spheres and projective and hyperbolic spaces),  allow some of the simplest
equivariant embeddings into a certain (pseudo) Euclidean space
E^N_(K) of suitable Hermitian matrices over an appropriate field 𝔽 by the projection
operators (see, for example <cit.>, <cit.>, <cit.> <cit.>, <cit.>, <cit.>). In the case of 𝔽P^m, this is the so-called first standard embedding, achieved, up to a rigid motion,  by the basis of eigenfunctions
corresponding to the first nonzero eigenvalue of the Laplacian. We use the notation ℍQ^m := ℍQ^m(4c),   c = ± 1, 
to denote jointly either of the two simply-connected quaternionic model space forms: the quaternionic projective space ℍP^m(4) or the quaternionic hyperbolic space ℍH^m(-4) of the respective quaternionic sectional curvatures 4 and -4.  We shall assume m ≥ 2.
Further, denote by  Φ : ℍQ^m →
E^N_(K) the embedding that associates to every quaternionic line in ℍ^m+1 (or time-like quaternionic line in 
the hyperbolic case) the operator (i.e. its matrix) of the
orthogonal projection onto it. The study of finite-type
submanifolds x: M^n →ℍQ^m is then the study of the
spectral behavior of the associated immersion x = Φ∘ x of M^n into E^N_(K), i.e. of the possibility of
decomposing x into finitely many eigenfunctions of Δ_M. It is therefore interesting to investigate to what extent the
analytic information contained in the spectral resolution (<ref>) of the immersion into finitely many terms, determines the geometry
of the submanifold.
A k-type immersion x satisfies
a polynomial equation in the Laplacian, P(Δ )(x - x_0) = 0,
where P(t) is a  monic polynomial of the least degree (k) with
coefficients that  are the elementary symmetric functions of the
eigenvalues λ_i,   i = 1, ⋯ , k <cit.>. Thus,
in the study of k-type submanifolds one necessarily deals with
the k-th iterated Laplacian Δ ^k x of the immersion,
which makes the investigation computationally very complex for
higher values of k. Hence, the most promissing study involves 
submanifolds of low type: 1, 2, or 3. In particular, immersions x: M^n → E_(K)^N which are of 2-type  satisfy 
Δ^2 x - a Δ x + b(x - x_0) = 0 for some constants a and b and their study is related to the study of (extrinsically) biharmonic 
submanifolds satisfying the condition Δ^2 x = 0.
Regarding the situation in complex space forms, the study of 1-type
submanifolds of ℂP^m was begun in <cit.> and a
parallel investigation for hypersurfaces of ℂH^m was
carried out in <cit.>. A complete classification of 1-type
submanifolds of a non-flat complex space form ℂQ^m (4c)
was achieved in our paper <cit.>. Submanifolds of Chen-type 2 have been
studied by several authors. Compact Kähler submanifolds of
ℂP^m were characterized by A. Ros <cit.> and
subsequently classified by Udagawa <cit.>, <cit.>, as compact
Einstein-Kähler parallel submanifolds, an example being the complex
quadric Q^m-1⊂ℂP^m.
Minimal surfaces of 2-type were classified by Shen
<cit.> and the classification of real hypersurfaces was undertaken
in <cit.> and  <cit.>, for hypersurfaces of
ℂP^m with constant mean curvature, and in <cit.>
for Hopf hypersurfaces of ℂQ^m. Those 2-type Hopf (or CMC) hypersurfaces in ℂP^m(4) include all geodesic spheres except 
the one which is of 1-type, two series of tubes about canonically embedded (totally geodesic) 
ℂP^k(4) ⊂ℂP^m(4) for k = 1, ..., m-2, and two particular tubes about the complex quadric Q^m-1. In a complex hyperbolic space the 
list of 2-type Hopf hypersurfaces includes only geodesic spheres and tubes about totally geodesic complex hyperbolic hyperplanes.
Submanifolds of non-flat real space forms which are of low type in a suitable (pseudo) Euclidean space of matrices via the immersion 
by projectors have been studied by several authors, see <cit.> and references there.

0.3truecm
Among other rank-1 symmetric spaces, results on finite-type submanifolds of quaternionic space forms
and Cayley planes are rare and those are not studied as extensively in these ambient spaces because of a more complex expression for the curvature and higher dimension of the distribution 𝒟^⊥.
1-Type submanifolds of quaternionic projective spaces were investigated
in <cit.> where quaternion CR and anti-CR submanifolds of 1-type were classified. In particular, the only 1-type hypersurfaces 
of ℍP^m (up to rigid motions) are open portions of a geodesic  hypersphere of radius r = ^-1√(3/(4m+1)). 
As a proper generalization of Hopf hypersurfaces of complex space forms to the quaternionic setting we have the notion of curvature-adapted
hypersurfaces of ℍQ^m, for which the normal Jacobi operator commutes with the shape operator, <cit.>, <cit.>, <cit.>. They are 
characterized by an equivalent condition that the maximal quaternionic subbundle 𝒟 of the tangent space is an invariant subspace 
of the shape operator (and thus its orthogonal
complement 𝒟^⊥ in TM is also invariant) <cit.>. 
In this paper we study curvature-adapted hypersurfaces of 
ℍQ^m of Chen-type 2, extending the results obtained for 2-type Hopf hypersurfaces in complex space forms to curvature-adapted ones in quaternionic space forms. Namely, we classify curvature adapted hypersurfaces of ℍP^m and ℍH^m (in the latter case assuming also constant principal curvatures) that are of 2-type in a suitably defined (pseudo) Eucldean space E^N_(K) of Hermitian matrices. They include geodesic hyperspheres of arbitrary radius except one, tubes about ℍH^m-1 in hyperbolic case, two series of tubes about quaternion projective spaces of lower dimension in ℍP^m, and two particular tubes about canonically embedded ℂP^m ⊂ℍP^m, whereas the other tubes about ℂP^m are shown to be of 3-type. Among these 2-type hypersurfaces we determine those that are mass-symmetric or minimal. We indicate how spectral decomposition can be obtained for a mass-symmetric 2-type hypersurface and give an explicit spectral resolution of x for geodesic spheres and tubes about ℂP^m(4) of radius r = 1/2^-1(1/√(m)) in ℍP^m(4). For a horosphere in ℍH^m we prove that it is not of finite type but satisfies the condition Δ^2 x = const≠ 0 and we prove that there are no real hypersurfaces of type 1 in ℍH^m. These results are given in Theorems 1-4 and Lemmas 2 and 6.
For the basic setup of the situation in quaternionic space forms and notation we refer to <cit.>, <cit.>, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, and <cit.>.

0.8truecm


§ PRELIMINARIES


0.8truecm

Let ℍ^m denote the m-dimensional quaternion number space (m ≥ 2), considered as a left vector space over (noncommutative) scalar  field 
of quaternions ℍ and let ℍQ^m:= ℍQ^m(4c) denote a m-dimensional non-flat model quaternionic space form, that is, either 
the quaternionic projective space ℍP^m(4) or the quaternionic hyperbolic space ℍH^m(-4) of constant quaternionic 
sectional curvature 4c = ± 4. 
As is well known, M:= ℍQ^m is a quaternion-Kähler manifold equipped with a quaternion-Kähler structure 
𝒥, which is a rank-3 vector subbundle of End  (TM) as oriented Riemannian bundle given an appropriate bundle metric and satisfying the following: For each p ∈M
there exists an open neighborhood G of p and local basis of sections J_1, J_2, J_3 ∈Γ (𝒥) of 𝒥
over G such that for every q ∈{1, 2, 3 } the following hold

    J_q^2 = - I,   and   J_q X, Y = -  X, J_q Y  ,

i.e. J_q is an almost Hermitian skew symmetric endomorphism of TM|_G for q = 1, 2, 3.

    J_q J_q+1 = J_q+2 = - J_q+1 J_q    (indices 3),


    ∇_X J ∈Γ (𝒥),   for all   X ∈Γ (TM),   J ∈Γ (𝒥),

i.e. 𝒥 is a parallel subbundle of End  (TM). Here and in what follows 
Γ is used to denote the set of all (local) smooth sections of a bundle.

Any triple {J_1, J_2, J_3 } of local sections of 𝒥 satisfying (<ref>)-(<ref>) is called a canonical local basis of
𝒥. As a consequence of these conditions there exist 1-forms θ_1, θ_2, θ_3 on G such that

    ∇_X J_q = θ_q+2(X) J_q+1 - θ_q+1(X) J_q+2   (indices 3 ),

for all X ∈Γ (TM),   q ∈{1, 2, 3 }. The Riemannian curvature tensor R of ℍQ^m(4c)
has the form

    R(X, Y)Z = c  [  Y, Z X -  X, Z Y + ∑_q=1^3(  J_qY , Z J_qX -
     J_qX , Z J_qY - 2  J_qX , Y J_qZ )   ],

for an arbitrary canonical local basis {J_1, J_2, J_3 } of 𝒥.
By using a particular (pseudo) Riemannian submersion one can construct ℍQ^m and its embedding Φ into a suitable
(pseudo) Euclidean space of matrices by means of projections to quaternionic lines. We refer to <cit.> where this construction is 
carried out in detail for ℍP^m 
and we note that a similar construction works, mutatis mutandis, also for ℍH^m since it is the same kind of construction used
in the complex setting for ℂQ^m, <cit.>, <cit.>, <cit.>. For ℍH^m setting see also <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.
Consider first the standard Hermitian form Ψ_c on ℍ^m+1 given by 
Ψ_c (z, w) = c  z_0 w_0 + ∑_j=1^m  z_j w_j,   z, w ∈ℍ^m+1,
with the associated (pseudo) Riemannian metric g_c = Re Ψ_c and the quadric hypersurface
N^4m+3:= { z ∈ℍ^m+1  |  Ψ_c (z, z) = c}. When c = 1,   N^4m+3 is the ordinary hypersphere S^4m+3 
of ℍ^m+1 = ℝ^4m+4 and when c = -1,   N^4m+3 is an indefinite hyperbolic space H_3^4m + 3 of index 3 in 
ℍ^m+1_1. The orbit space under the natural action of the group Sp(1) = S^3 of unit quaternions
on N^4m+3 defines ℍQ^m(4c), which becomes the base space of a (pseudo) Riemannian submersion with totally geodesic fibers. 
The standard embedding Φ into the set of Ψ_c-Hermitian matrices 𝐇^(1)(m+1) is achieved by identifying  a point, that is a 
quaternion line (or a time-like quaternion line in the hyperbolic case) with the projection operator onto it. Then one gets the 
following matrix representation of Φ at a point p = [z], where z = (z_j) ∈ N^4m+3⊂ℍ^m+1_(1)is a row vector:

    Φ ([z]) = [     |z_0|^2  c z̅_0 z_1           ⋯  c z̅_0 z_m;   z̅_1  z_0   c |z_1|^2           ⋯  c z̅_1 z_m;           ⋮           ⋮           ⋱           ⋮;   z̅_m  z_0 c z̅_m  z_1           ⋯   c |z_m|^2 ] .

The image Φ (ℍQ^m) of the space form considered is the set {P} of Hermitian projectors (P^2 = P) with trace 1 and 
is contained in the hyperquadric of 𝐇^(1)(m+1) centered at I/(m+1) and defined by the equation 

    ⟨ P - I/(m+1),  P - I/(m+1)⟩ = cm/2(m+1),

where I denotes the (m+1)× (m+1) identity matrix and the metric on 𝐇^(1)(m+1) is given by ⟨ S, T⟩ = c/2  Re tr (S T).
The second fundamental form  of this embedding is parallel, i.e. ∇σ = 0.
The following formulas for the shape operator of Φ in the direction of σ (X,Y) follow from <cit.> and 
are extensions of the corresponding well-known formulas of A. Ros in the complex case (see, for example <cit.>, <cit.>, <cit.>, <cit.>).

    (X,Y),  (V, W)    =    c  [2  X, Y V, W  +
      X, V Y, W  +  X, W Y, V 
                + ∑_q=1^3 (  J_qX, V J_qY, W  +  J_qX, W  J_qY, V ) ],



    A_σ (X,Y)V = c  [2 ⟨ X, Y⟩ V + ⟨ X,
    V⟩ Y + ⟨ Y, V⟩ X + ∑_q=1^3 (⟨ J_qX, V⟩ J_qY + ⟨
    J_qY, V⟩ J_qX )].


One also verifies (see <cit.>)

    ⟨σ (X, Y), x⟩ = - ⟨ X, Y
    ⟩ ,     ⟨σ (X, Y),   I ⟩ =   0,     σ (J_qX, J_qY) = σ (X, Y),   q = 1, 2, 3.


For additional properties of the embedding Φ see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.


Let x: M^n →ℍQ^m be an isometric immersion of a Riemannian n-manifold as a real hypersurface of a quaternionic 
space form (thus n = 4m-1). Then we have the associated composite immersion x = Φ∘ x, which realizes M as a submanifold 
of the (pseudo) Euclidean space E^N_(K):= 𝐇^(1)(m+1) of dimension N = (m+1)(2m+1) and appropriate index K, equipped with the 
usual trace metric A, B  = c/2 Re tr  (AB). In this notation the subscripts and superscripts in parenthesis 
are present only in relation to ℍH^m, so that the superscript 1 in 𝐇^(1)(m+1) is optional and appears only in 
the hyperbolic case, since the construction of the embedding is based on the form Ψ in ℍ_1^m+1 of index 1.
Let ξ be a local unit vector field normal to M in ℍQ^m ,  A the shape operator of the
immersion x, and let κ = (1/n) tr  A be the mean curvature of M in ℍQ^m, so that the mean
curvature vector H of the immersion equals H = κξ .
Further, let ∇,  A, D, denote respectively the Levi-Civita
connection, the shape operator, and the metric connection in the normal
bundle, related to ℍQ^m and the embedding Φ. Let the same letters
without bar denote the respective objects for a submanifold M and the immersion x,
whereas we use the same symbols with tilde to denote the corresponding objects related to the
composite immersion x̃ : = Φ∘ x of M into the (pseudo) Euclidean space
𝐇^(1)(m+1). As usual, we use  for the second fundamental form of
ℍQ^m in E^N_(K) via Φ and h for the second fundamental form of
a submanifold M in ℍQ^m. An orthonormal basis of the tangent space T_pM at a general
point will be denoted by {e_i }, i = 1, 2, … , n, and indices i, j will generally range from 1 to n, whereas q and k range from 1 to 3. 
We give first some important formulas which will be repeatedly used throughout this paper.
For a general submanifold M, of a Riemanian manifold M, local tangent fields X, Y ∈Γ (TM) and a
local normal field ξ∈Γ (T^⊥ M), the
formulas of Gauss and Weingarten are

    ∇_XY = ∇_XY +
    h (X, Y) ;         ∇_X ξ = -A_ξ X + D_Xξ.


In particular, for a hypersurface of a quaternionic space form ℍQ^m
with (locally defined) unit normal vector field ξ and the corresponding shape operator A = A_ξ, they
become

    ∇_XY =∇_XY + ⟨ AX, Y ⟩ξ
    ;        ∇_Xξ = - AX.
 

Let {J_q},   1 ≤ q ≤ 3, be a triple of  almost complex structures of ℍQ^m which form a canonical basis of 𝒥 
and U_q ∈Γ (TM) be the tangent vector fields defined by U_q:= -J_qξ ,   q = 1, 2, 3. Define distributions 𝒟^⊥ = Span_ℝ{U_1, U_2, U_3  } and 𝒟 to be the orthogonal complement of 𝒟^⊥ in TM so that 𝒟 is the maximal subbundle of TM which is left invariant by the quaternionic structure 𝒥|_TM. 
Further, define endomorphisms
S_q of the tangent space and a normal bundle valued 1-forms F_q by

    S_qX = (J_qX)_T ,    F_qX = (J_qX)_N  = ⟨ X, U_q ⟩ξ ,

i.e for X ∈Γ (TM),    J_qX = S_qX + F_qX is the
decomposition of J_qX into tangential and normal to submanifold parts. Further, we obtain the following from (<ref>) - (<ref>) and (<ref>):

    S_qU_q = 0,     S_q U_q+1 = U_q+2,    S_q U_q+2 = - U_q+1



    S_qX = J_qX - ⟨ X, U_q ⟩ξ ,      S_q^2X = -X + ⟨ X, U_q ⟩ U_q,



    ∇_XU_q = S_qAX + θ_q+2(X)  U_q+1 - θ_q+1(X)  U_q+2,


Note that both 𝒟 and 𝒟^⊥ are S_q-invariant, for each q = 1, 2, 3. The equations of Codazzi for a hypersurface of ℍQ^m(4c) is  given by

    (∇ _XA)Y - (∇_YA)X = c ∑_q=1^3 [⟨ X, U_q ⟩ S_qY -
    ⟨ Y, U_q ⟩ S_qX - 2⟨ S_qX, Y ⟩ U_q].



For any hypersurface M of a Riemannian manifold M with curvature tensor R one defines the normal
Jacobi operator 𝒦∈End  (T_pM) at a point p ∈ M by 𝒦(X):= R (X, ξ ) ξ where ξ is
a unit normal to M at p (determined up to a sign) and X an arbitrary tangent vector at p. A hypersurface M is said to be 
curvature-adapted to M if the normal Jacobi operator 𝒦 of M commutes with the shape operator A at every point, i.e. the two operators are simultaneously diagonalizable. Equivalently, 
𝒦∘ A = A∘𝒦 translates into R (AX, ξ ) ξ = A (R (X, ξ ) ξ ), 
for every X ∈Γ (TM), and for a hypersurface of a quaternionic space form 
ℍQ^m (4c) of quaternionic sectional curvature 4c whose curvature tensor is given by (<ref>) this 
further leads to

    ∑_q = 1^3 ⟨ AX, U_q⟩ U_q = ∑_q = 1^3 ⟨ X, U_q⟩ AU_q,   for every   
    X ∈Γ (TM),
 
i.e. (AX)_𝒟^⊥ = A(X_𝒟^⊥), which in turn is equivalent to the condition that the distributions 
𝒟^⊥ and 𝒟 in TM are invariant under the action of the
shape operator A, meaning A 𝒟^⊥⊂𝒟^⊥ and A 𝒟⊂𝒟. The notion of curvature-adapted hypersurface was introduced in <cit.> and effectively used in the study of hypersurfaces in <cit.>, <cit.>, more recently in <cit.>, <cit.>, <cit.> and other works.
For such a hypersurface
AU_q = ∑_k=1^3 a_q^k U_k for some real functions a_q^k, symmetric in k and q, and then using
(<ref>) we get ∑_q S_q A U_q = 0.

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The gradient of a smooth function f is a vector field
∇ f := ∑_i (e_i f)e_i and the Laplacian acting on smooth functions is defined as 
f := ∑_i [(∇_e_ie_i) f - e_i e_i f], where { e_i}, 1 ≤ i ≤ n, is an orthonormal basis of the tangent space of M.
The Laplace operator can be extended to act on a vector field V in E^N_(K) = 𝐇^(1)(m+1) along x
(M) by

    Δ V = ∑_i [∇_∇_e_ie_i V - ∇_e_i∇_e_i V].

The product formula for the Laplacian, which will be used in the ensuing
computations, reads

    Δ (f  g) = (Δ f)  g + f(Δ g) - 2 ∑_i (e_if)( e_ig) ,

for smooth functions f,  g ∈ C^∞ (M) and it can then be extended to hold for the scalar
product of vector valued functions, hence also for product of matrices, in a natural way.  We shall use
the notation f_k:= tr  A^k, and in particular f:= f_1 = tr  A.
For an endomorphism B of the tangent space of M we define
tr (∇ B): = ∑_i=1^n (∇_e_iB)e_i . We will use V_μ to denote vector space of principal vectors (eigenvectors of A) corresponding to principal curvature μ and s(𝒟),  s(𝒟^⊥) to denote the spectrum (the set of eigenvalues) of A when restricted to 𝒟 and 𝒟^⊥, respectively, at a point considered.
We shall assume all manifolds to be smooth and connected, and all immersions smooth.
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§ THE SECOND ITERATED LAPLACIANS OF A REAL HYPERSURFACE OF ℍQ^M


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From a well-known formula of Beltrami we have 

    Δx = - n H = - f ξ - ∑_i=1^n σ (e_i, e_i ),

where here, and in the following, we understand the Laplacian Δ of M to be applied to
vector fields along M (viewed as E^N_(K)-valued functions, i.e.  matrices) componentwise.

The product formula (<ref>) gives


    Δ^2x := Δ (Δx ) = - (Δ f )ξ - f
    (Δξ ) + 2 σ (∇ f , ξ ) - 2 A(∇ f )- ∑_i
    Δ  (σ (e_i, e_i )).

Moreover, we compute

    Δξ   =   ∑_i [∇_∇_e_ie_iξ -
    ∇_e_i∇_e_iξ ]
       =   ∑ _i [-A(∇_e_ie_i ) + σ (∇_e_ie_i, ξ ) + ∇_e_i(Ae_i) 
    + σ (e_i, Ae_i) + A_σ (e_i, ξ ) e_i - D_e_i (σ (e_i, ξ ))].
 
Using (<ref>), the parallelism of σ ,  and the fact that tr  (∇ A) = ∑_i = 1^n (∇_e_iA) e_i =  ∇ (tr  A) = ∇ f
(which follows from the Codazzi equation), we obtain

    Δξ = ∇ f + [f_2 + c(n - 3)] ξ - f σ (ξ, ξ ) + 2 ∑_i σ (e_i, Ae_i).


Further computations yield

    ∑_i ∇_X (σ (e_i, e_i)) = - 2 c (n + 4) X + 2 c ∑_q = 1^3 ⟨ X, U_q ⟩ U_q + 2 σ (AX, ξ),

so that starting from (<ref>) and using (<ref>) - (<ref>) and (<ref>) we get


    ∑_i Δ ( σ (e_i, e_i))    =    2c   [ (n+5)f - 2 ∑_q=1^3 ⟨ AU_q, U_q ⟩ ] ξ 
    -  4c ∑_q=1^3 S_qAU_q - 2(3c + f_2 ) σ (ξ, ξ )
               - 2 σ (ξ, ∇ f) +  2c  (n+4) ∑ _i σ (e_i, e_i ) + 2 ∑_i σ (Ae_i, Ae_i).
 

Combining formulas (<ref>) - (<ref>) we finally obtain 


    Δ^2 x̃   =    - [Δ f + ff_ 2 + c( 3n + 7)f - 4c ∑_q=1^3 ⟨ AU_q, U_q ⟩ ] ξ 
    - f ∇ f - 2 A (∇ f) 
             +   4c ∑_q=1^3 S_qAU_q  + ( 6c + 2 f_2 + f^2 ) σ (ξ, ξ ) + 4 σ (∇ f , ξ )
             -   2 c(n+4) ∑_i σ (e_i, e_i ) - 2 f ∑_i σ (e_i, Ae_i ) -  2 ∑_iσ (Ae_i, Ae_i ) ,
 
which holds for any real hypersurface of ℍQ^m(4c).
Compare this with a similar formula in <cit.>, <cit.> and formula (2.9) of <cit.>.
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§ CURVATURE-ADAPTED HYPERSURFACES OF 2-TYPE WITH CONSTANT PRINCIPAL CURVATURES 

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 In this section we work with 2-type curvature adapted hypersurfaces of ℍQ^m that have constant principal curvatures.
As a matter of fact, it was proved in <cit.>  that a curvature-adapted hypersurface of ℍP^m necessarily has constant principal curvatures
without any additional assumptions, but that is not yet known for hypersurfaces of ℍH^m.
Assume that a general hypersurface M^n ⊂ℍQ^m ,   n = 4m-1, is of Chen-type 2 via the embedding Φ, i.e. 
x = x_0 + x_u + x_v according to (<ref>). Then by taking succesive Laplacians and eliminating 
x_u , x_v we get

    Δ^2 x - a Δx + b x = b x_0 ,   with   
    a := λ_u + λ_v,    b := λ_u λ_v

Conversely, (<ref>) implies that M^n is of type ≤ 2 when M is compact or when the trinomial P(t) = t^2 - a t + b has two 
distinct real roots <cit.>. Denote the vector field along x (M) represented by the left-hand side of (<ref>) by L. 
Then, differentiating L with respect to an arbitrary tangent vector field X and taking the inner product with x, 
using (<ref>), (<ref>) and (<ref>), gives

    0    =   ⟨∇_X L, x⟩ = X ⟨ L, x⟩ - ⟨ L, X ⟩
       =    X (f^2 - 6c + 2cn (n+4) - an + bc/2) + ⟨ f ∇ f + 2 A (∇ f), X ⟩ - 4c ∑_q = 1^3 ⟨ S_q A U_q, X ⟩ ,

from where we obtain

    2 A (∇ f) + 3 f ∇ f = 4c ∑_q=1^3  S_q A U_q .

 Consequently, (<ref>) yields 

    A (∇ f ) = - 3f/2∇ f ,

for a 2-type curvature-adapted hypersurface. Thus, on an open (possibly empty) set {∇ f ≠ 0 } the gradient of f is a principal direction. Other conditions can be obtained by considering various components of differentiated 2-type equation (<ref>), but the expressions are rather complicated compared to ℂQ^m (4c) setting. Although those kinds of computations were instrumental in proving that a 2-type Hopf hypersurface of ℂQ^m has constant mean curvature and, moreover, constant principal curvatures, those conclusions do not follow in a similar way in quaternionic setting. Hence, for the rest of this paper, we consider now curvature adapted hypersurfaces of type 2 that have constant principal curvatures. This is not any restriction in ℍP^m case, since a curvature-adapted hypersurface of ℍP^m has (locally) constant principal curvatures and full classification of hypersurfaces with constant principal curvatures in both ℍP^m and ℍH^m is known <cit.>.
The information on these model curvature-adapted hypersurfaces of ℍQ^m(4c)  with constant principal curvaturess in terms of principal curvatures μ, ν,  α_i,   and their multiplicities m(μ), m (ν),   m(α_i),   i = 1, 2,  is presented in  Table 1 below taken from <cit.>. The left portion of the table corresponds to the projective case and the right portion to the hyperbolic one. Here
μ and ν (resp. α_1 and α_2) are principal curvatures of A|_𝒟 (resp. A|_𝒟^⊥).
These hypersurfaces are usually categorized as hypersurfaces of class (type) A and class (type) B. Class-A hypersurfaces are P_1^k(r) in ℍP^m(4) which form a family of tubes of some radius r ∈ (0, π/2) about a canonically embedded ℍP^k(4), and H_1^k(r) in ℍH^m(- 4), which form a family of tubes of some radius r > 0 around a canonically embedded ℍH^k(- 4) for some k ∈{0, 1, … , m - 1  }. A particular subclass A_1 is obtained when k = 0, producing a family of geodesic hyperspheres P_1^0(r) and H_1^0(r), or when k = m - 1 in the hyperbolic case, producing a family of equidistant hypersurfaces to canonically imbedded (totally geodesic) hyperplane ℍH^m - 1(- 4). Class-A_2 hypersurfaces are all the other tubes P_1^k(r),   H_1^k(r),   k ≠ 0,  m - 1. The horosphere H_3 in ℍH^m(- 4) is of class A_0. Class-B hypersurfaces are P_2(r) in ℍP^m(4) which are tubes of some radius r ∈ (0, π/4) about a canonically embedded complex projective space ℂP^m(4) of half the dimension and H_2(r) in ℍH^m(- 4) which are tubes of some radius r > 0 about a canonically embedded complex hyperbolic space ℂH^m(- 4).
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In a neighborhood of every point of some open dense set 𝒲 of curvature-adapted hypersurface M^n according to Lemma 3.6 of 
<cit.> we can choose a local canonical 
basis {J_1, J_2, J_3 } of quaternionic bundle 𝒥 over some open neighborhood G of ℍ Q^m  and a local unit normal field of M defined on G: = G∩ M ⊂𝒲 so that U_q: = - J_q ξ ,    q = 1, 2, 3 are principal on G. Let α_q be 
the corresponding (locally constant) principal curvatures, i.e. A (U_q) = α_q U_q and let s (𝒟)  and s (𝒟^⊥) denote the spectrum (collection of eigenvalues) of A|_𝒟 and A|_𝒟^⊥, respectively. 
Formula (<ref>) becomes

    Δ^2 x   =   { 4c ∑_q=1^3 α_q - f  [ f_ 2 + c( 3n + 7)] } ξ + 
    ( 6c + 2 f_2 + f^2 ) σ (ξ, ξ ) 
               - 2 c(n+4) ∑_i σ (e_i, e_i ) - 2 f ∑_i σ (e_i, Ae_i ) -  2 ∑_iσ (Ae_i, Ae_i ) .
 

Let X ∈Γ (TM). Using formulas (<ref>), (<ref>), (<ref>), (<ref>) and the fact that  σ is parallel, we get from (<ref>)

    ∇_X (Δx ) = 2c (n + 4) X + f AX - 2 c ∑_q  X, U_q  U_q - f  (X, ξ ) - 2  (AX, ξ ),

where X_𝒟^⊥:= ∑_q  X, U_q  U_q is the component of the tangent vector X which belongs to 𝒟^⊥ .
Likewise, from (<ref>) we have


    ∇_X (^2 x )    =     [2c f^2 + 4 (n^2 + 8n + 13)] X 
    - { 4c ∑_q α_q - f [f_2 + c (3n + 11)]}  AX + 4c A^2 X  
               - 2c  [2 f_2 + f^2 + 2c (n + 7)]  X_𝒟^⊥ - 4cf ∑_q J_q AS_q X - 4c ∑_q J_q A^2 S_q X 
               + { 4c ∑_q α_q - f [f_2 + c (3n + 7)]}  (X, ξ )- 2  [2f_2 + f^2 + 2c (n + 7)]  (AX , ξ) 
               - 4 f  (A^2X, ξ ) - 4  (A^3X, ξ ) - 2 f ∑_i  ((∇_XA)e_i , e_i) -  4 ∑_i  ((∇_XA)e_i , Ae_i).




Differentiating (<ref>) with respect to X we get

    ∇_X (Δ^2 x) - a  ∇_X (x) + b X = 0.

Therefore, using the above expressions and separating the part of (<ref>) which is tangent to ℍQ^m we get

    [ 2c f^2 + 4 (n^2 + 8n + 13)  - 2c (n + 4) a + b ]   X + 4c  A^2X 
           - { 4c ∑_q α_q - f [f_2 + c (3n + 11) - a]}  AX - 2c  [2f_2 + f^2 + 2c (n + 7) - a ]  X_𝒟^⊥
           - 4cf ∑_q S_q A S_q X  - 4c ∑_q S_q A^2 S_qX = 0 ,

whereas the part normal to ℍQ^m yields

    { 4c∑_q α_q  - f [f_2  + c (3n + 7) - a]}   (X, ξ) - 4  (A^3X, ξ )
           - 4f  (A^2X, ξ ) - 2  [2f_2 + f^2 + 2c (n + 7) - a]   (AX, ξ ) 
           - 2 f ∑_i  ((∇_XA) e_i , e_i) - 4 ∑_i  ((∇_XA)e_i , Ae_i) = 0.


Note that J_q A^k S_q X has no ξ -component since U_q is principal and S_qU_q = 0. 
 
These expressions are linear in X. Further separation of parts relative to the splitting
𝒟⊕𝒟^⊥ of the tangent space yields the following

 Let M^n be a connected, curvature-adapted real hypersurface of 
ℍQ^m(4c) (m ≥ 2,   n = 2m-1). Suppose M has constant principal curvatures and α_q^2 + 4 c ≠ 0, for all q =1, 2, 3,  α_q ∈𝔰 (𝒟^⊥). If M is of 2-type via x 
satisfying a 2-type condition (<ref>) then the following relations hold:


  (C_1)  

    [2c (n + 3) + α_k f]  a    =     b + 4 (n + 1)(n + 6) +  α_k { f  [f_2 + c (3n + 7)] - 4 c ∑_q α_q }
              - 4c f_2 + 4c [ f  ∑_q α_q + ∑_q α_q^2 ],

for every α_k ∈𝔰 (𝒟^⊥),    k = 1, 2, 3;


  (C_2)   
    [2c (n + 4) + τ f]  a    =      b + 4 (n^2 + 8n + 13)  + 4c { f( τ + ∑_q τ_q ) +  ( τ^2 +  ∑_q τ_q^2 )}
              + 2cf^2 + { f   [f_2 + c (3n + 7)] - 4c ∑_qα_q }τ ,

for any principal curvature τ∈𝔰 (𝒟) ;


  (C_3)   
    (f + 2 τ )  a    =    f [f_2 + c (3n + 7)] - 4c∑_q α_q + 4 τ [f ( τ +   ∑_q τ _q ) + ( τ^2 + ∑_q τ _q^2 )] 
              +   2 τ [ 2 f_2 + f^2 + 2c (n + 7) - 2f ∑_qα_q - 2 ∑_q α_q^2] ,

for any τ∈𝔰 (𝒟) ;


  (C_4)   
    f  (∇_XA)Y , Z  + f ∑_q  (∇_XA)(S_qY) , S_qZ 
             +   (∇_XA^2)Y , Z  + ∑_q  (∇_XA^2)(S_qY) , S_qZ  = 0,

for every Y, Z ∈Γ (𝒟) and X ∈Γ(TM).

Conversely, if (C_1) - (C_4) hold for a curvature-adapted hypersurface of ℍQ^m with constant principal curvatures, 
where a and b are constants and τ∈𝔰 (𝒟) is an arbitrary principal curvature on 𝒟, then the 
formula (<ref>) holds and the submanifold is of finite type  ≤ 2 if the corresponding monic polynomial 
P(t) = t^2 - a t + b has two distinct real roots.

The condition α_q^2 + 4 c ≠ 0 for all q = 1, 2, 3 is one case considered by Berndt in his analysis, under which for every τ∈𝔰_p(𝒟) at every 
point p and for every q = 1, 2, 3, there exists exactly one τ_q ∈𝔰_p(𝒟) such that S_q V_τ = V_τ_q
and τ_q satisfies 

    [2 τ - α_q]  [2 τ_q - α_q] = α_q^2 + 4c,

 according to Lemma 4.9  of <cit.>. Those are the τ_q's appearing in conditions (C_2) and (C_3). The complementary case α_q^2 + 4 c = 0 holding for some q = 1, 2, 3 leads to α_1 = α_2 = α_3 =: α. Since it is either proved or assumed in <cit.> that the principal curvatures are constant, we find from Table 1 that the only hypersurface with constant principal curvatures for which α_q^2 + 4 c = 0 holds is the horosphere H_3, which is not of 2-type as will be proved in Lemma 2 below and this case can be excluded from further discussion.


Assume that the cannonical triple of quaternionc sections J_1, J_2, J_3 has been chosen on an open neighborhood G_p of an arbitrary point p of the dense set 𝒲⊂ M, so that U_q,   q = 1, 2, 3, are principal directions. This will be the standing assumption in all subsequent calculations. By linearity of the above formulas in X it suffices to consider various components of (<ref>) and (<ref>) when X belongs to
all possible eigenspaces of A. Note first that by (<ref>) we have

    ∑_q S_q A S_q U_k = (α_k - ∑_q α_q ) U_k,   ∑_q S_q A^2 S_q U_k = (α_k^2 - ∑_q α_q^2 ) U_k,     k = 1, 2, 3.

Then (C_1) follows from (<ref>) when we set X = U_k for k = 1, 2, 3, and use (<ref>). Next, consider X ∈ V_τ to be any principal direction of A|_𝒟 for a principal curvature 
τ∈𝔰 (𝒟). 
Since in this case A S_q X  = τ_q   S_q X we get

    ∑_q S_q A S_q X = - (∑_q τ_q) X,   ∑_q S_q A^2 S_q X = - (∑_q τ_q ^2) X.

Taking these into consideration, choose such an X in (<ref>) to get (C_2).
Regarding the components that come from the part normal to ℍQ^m, we recal that the normal space of ℍQ^m in 
H^(1)(m+1) = E^N_(K) is spanned by x and the values of  on various pairs of tangent vectors of ℍQ^m, namely by
(ξ , ξ), (ξ , X), (ξ , W),   (X, Y),   (X, W), and (V, W) for X, Y ∈Γ (𝒟), 
V, W ∈Γ (𝒟^⊥ ) <cit.>. Equation (<ref>) and hence equation (<ref>) have no x-component since 
(<ref>) holds and f and f_2 are constant. Note also that by (<ref>)
(ξ , U_q) = 0,   (U_q, U_q) =  (ξ , ξ) and (U_q, X) =  (ξ , J_qX) for X ∈𝒟. 
Conditions (C_3) and (C_4) follow from the normal part (<ref>). Namely, let ℒ denote the left-hand side of (<ref>). We consider the metric product of ℒ with normal vectors of the above form.

By using (<ref>) - (<ref>),  the fact that f, f_2 are constant, and U_q, q = 1, 2, 3 are principal, it is checked in a straightforward manner that ⟨ℒ,  σ(ξ , ξ ) ⟩ = 0 so σ (ξ , ξ)-component
gives no additional information. By (<ref>) and symmetry of σ we have

    σ (ξ , U_q) = σ (J_qξ , J_qU_q) = - σ (ξ , U_q) = 0,

thus by linearity σ (ξ , W) = 0 for every W ∈Γ (𝒟^⊥) and there is no need to consider components of this form.

Now take Y ∈Γ (𝒟). We compute (ξ , X ),  (ξ , Y )    = c  X, Y and from (<ref>) and (<ref>)

    ∑_i  (ξ , Y ),  ((∇_XA)e_i, e_i)   = -2c (∑_q α_q ) AX, Y  - 2 c ∑_q  S_qAS_qAX, Y  ,



    ∑_i ⟨σ (ξ , Y ) , σ ((∇_XA)e_i, Ae_i) ⟩  = - c (∑_q α_q^2 )⟨ AX, Y⟩  -  c ∑_q ⟨ S_qA^2S_qAX, Y ⟩ .

Therefore, the inner product of (<ref>) with (ξ , Y ) gives

    4 ∑_q  S_qA^2S_qAX, Y  + 4f ∑_q S_qAS_qAX , Y  - 4  A^3X, Y  - 4f  A^2X, Y
           -2  [2 f_2 + f^2 + 2c (n + 7) - a - 2f ∑_q α_q - 2 ∑_q α_q^2]  AX, Y
           + {4c ∑_q α_q + f a - f  [f_2 + c (3n + 7)] } X, Y = 0.

Since  A𝒟⊂𝒟,   S_q 𝒟 = 𝒟, and  the expression
is linear in X, Y ∈𝒟 we can drop Y and take X ∈ V_τ⊂𝒟 for any τ∈𝔰 (𝒟)
using AX = τ X ,   A(S_qX) = τ_q S_qX, with τ , τ _q related by (<ref>), to get (C_3). 
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Finally, to prove (C_4), first observe that there is no need to consider (W, Y)-component of ℒ for W ∈𝒟^⊥,   Y ∈𝒟 since 
W = ∑_q c_q U_q for some scalars c_q and thus

    (W, Y) = ∑_q c_q  (U_q, Y) = ∑_q c_q  (ξ , J_qY),

with J_qY belonging also to Γ (𝒟) and we alredy considered components of the form (ξ , 𝒟 ). We also know 
that (W, ξ ) = 0. Moreover, for V, W ∈𝒟^⊥ we have V = ∑_j c_j U_j,   W = ∑_k d_k U_k,   j, k = 1, 2, 3 and

    (V, W) = ∑_k=1^3 c_k d_k  (U_k, U_k) + ∑_k=1^3∑_j≠ k c_j d_k  (U_j, U_k) = (∑_k=1^3 c_k d_k)  (ξ , ξ),

the component already considered. Therefore, there remain only components of the form (Y, Z),   Y, Z∈Γ (𝒟) to be 
considered. Since Y, Z ∈𝒟, by (<ref>) we have (X, ξ ),  (Y, Z) = 0 for any X ∈Γ (TM). Further, since f and f_2 are constant

    ∑_i  ((∇_XA)e_i ,  e_i),  (Y, Z)  = 2c   (∇_XA)Y,  Z   + 2c ∑_q  (∇_XA)(S_qY), S_qZ


    ∑_i  ((∇_XA)e_i ,  A e_i),  (Y, Z)  = c   (∇_XA^2)Y,  Z   + c ∑_q  (∇_XA^2)(S_qY), S_qZ  .


As the last condition, for the component in σ(Y, Z) direction we look at  ⟨ℒ, σ (Y , Z) ⟩ = 0, where ℒ is as above with X ∈Γ (TM) and Y, Z ∈Γ (𝒟) Using the preceding calculations and (<ref>),  this is easily shown to be equivalent to (C_4).
Conversely, since we considered all possible components, the conditions (C_1) - (C_4) are equivalent to (<ref>) and (<ref>) 
by linearity and thus we get (<ref>), from which it follows that a hypersurface is of type ≤ 2, provided that the 
corresponding polynomial has two distinct real roots.


Note that by the results of Pak <cit.> for the projective case and Lyu, Pérez and Suh <cit.> for the hyperbolic case (see also <cit.>), any of the class-A hypersurfaces in ℍQ^m from either list  is
characterized by

    (∇_XA)Y = -   c ∑_q = 1^3  [ S_q X, Y  U_q +  U_q, Y  S_qX],    X, Y ∈Γ (TM),

so that the condition (C_4) is trivially
satisfied for those hypersurfaces. Further, by eliminating b from (C_1) and (C_2) we get

    [2c +  (τ - α_k) f ]  a    =      4(n + 7) + 2 c f^2 + 4 c f_2  + 4c  {f ( τ +  ∑_q τ_q) + ( τ^2 + ∑_q τ_q ^2 )}
           + { f  [f_2 + c (3n + 7)] - 4 c ∑_q α_q }  ( τ - α_k) - 4 c [f ∑_q α_q + ∑_q α_q ^2  ],
 
for any k = 1, 2, 3 and τ∈𝔰 (𝒟). If a can be uniquely determined from this condition (regardless of the choice of τ and α_k and  consistent with (C_3), then b is
uniquely determined from (C_1).
0.7truecm


§ THE MAIN RESULTS 

0.6truecm
First, we show that a horosphere cannot be of 2-type and moreover

 
The horosphere H_3 in ℍH^m satisfies Δ^2x = const≠ 0 and is not of any finite type via x. 


 
From Table 1 we find the principal curvatures μ = 1,   m(μ) = n - 3 and α = 2,   m(α) = 3 so that f = n + 3,   f_2 = n + 9 and ∑_q α_q = 6. Hence, from (<ref>) with c = -1 we find


    Δ^2x  =   2    (n^2 + 2n - 15) ξ + (n^2 + 8 n + 21) σ (ξ , ξ)
        + 2 (n + 4) ∑_e_i ∈𝒟σ (e_i, e_i) - 2 (n + 3) ∑_e_i ∈𝒟σ (e_i, e_i) - 2 ∑_e_i ∈𝒟σ (e_i, e_i)
        + 2 (n + 4) ∑_q  σ (U_q, U_q) - 4 (n + 3)  ∑_q  σ (U_q, U_q) - 8 ∑_q  σ (U_q, U_q) 
    
    =       (n^2 + 2 n - 15)  [  2 ξ +  σ (ξ , ξ)  ]   ,

since the terms on the second line cancel out and the terms on the third line combine with σ(ξ ,ξ) term according to (<ref>). Hence, for X ∈Γ(TM) we have 

    ∇_X(Δ^2x)    = (n + 5)(n - 3)  [2 σ(X, ξ) - 2 AX - A_σ(ξ , ξ) X - 2 σ (AX , ξ)] 
       =   2 (n + 5)(n - 3)  [σ (X, ξ) -  σ (AX, ξ) - AX + X + X_𝒟^⊥]

When X ∈Γ(𝒟) then AX = X and this expression is equal to zero,  whereas when X ∈Γ (𝒟^⊥) then AX = 2 X so that

    ∇_X(Δ^2x) =  - 2  (n + 5)(n - 3) σ(X, ξ).


By (<ref>) and symmetry of σ we have  σ (U_q , ξ) = 0,   q = 1, 2, 3, so that σ (X, ξ) = 0 for X ∈Γ (𝒟^⊥), and hence, by linearity, valid for all tangent vectors X. Therefore, Δ^2x =: C is a constant vector (matrix), i.e. 2 ξ + σ (ξ , ξ) = const and this constant is nonzero since σ (ξ, ξ) ⊥ξ.
If we assume that H_3 is of finite k-type via x, then P(Δ)(x - x_0) = 0 where P is a monic polynomial of degree k. Since Δ^sx = 0 for s > 2 we would have C - a_1  Δx + a_0 (x - x_0) = 0 for some constants a_0, a_1 where a_1 ≠ 0 since x is not constant. Then ∇_X(Δx) - h X = 0, for X ∈Γ (TM) where h = a_0/a_1. By (<ref>) this would imply

    -   [2 (n + 4) +  h]  X + 2 X_𝒟^⊥ + (n + 3) AX - (n + 3)σ (X, ξ) - 2 σ (AX , ξ) = 0,

for any finite type k ≥ 1.
Putting X ∈𝒟,   AX = X and X ∈𝒟^⊥,   AX = 2 X into this formula and separating parts tangent to M, we get h = - (n +5) and h = 0, respectively, which is a contradiction, proving that H_3 cannot be of k-type for any finite k. 


 
There exists no real hypersurface of    ℍH^m(- 4),   m ≥ 2, which is of Chen type 1 in H^1(m + 1) via Φ. Here, we do not assume hypersurface to be curvature-adapted.


   
Suppose that a hypersurface M ⊂ℍH^m(-4) is of Chen-type 1 so that

    x = x_0 + x_u,   with  x_0 = const  and  Δx_u = λx_u

according to (<ref>). Then Δx - λx + λx_0 = 0. Differentiating this relation with respect to X ∈Γ (TM) we get ∇_X (Δx) - λ X = 0, which yields, by way of (<ref>) and (<ref>),

    - (X f) ξ - f σ(X, ξ) - 2 σ (AX , ξ) + f AX + 2 c (n + 4) X - 2 c X_𝒟^⊥ - λ X = 0 .

Separating parts that are tangent to M, normal to M but tangent to ℍH^m and normal to ℍH^m we get first from the ξ-component that f = tr A is constant and 

    f AX + 2c (n + 4) X - 2 c X_𝒟^⊥ - λ X = 0,


    2 σ (AX , ξ) + f σ (X, ξ) = 0.

Choosing X in (<ref>) to belong to 𝒟 and 𝒟^⊥ we get, respectively,

    f AX + [2 c (n + 4) - λ] X = 0  for   X ∈𝒟  and    f AX + [2 c (n + 3) - λ] X = 0  for   X ∈𝒟^⊥.

This implies f ≠ 0 and in our case c = - 1, so

    AX = [λ + 2 (n + 4)]/f X   for   X ∈𝒟  and     AX = [λ + 2 (n + 3)]/f X  for   X ∈𝒟^⊥.

Therefore, there are two constant principal curvatures

    1/f  [λ + 2 (n + 4)]    and  1/f  [λ + 2 (n + 3)]

of respective multiplicities n - 3 and 3. Since A𝒟⊂𝒟,  A𝒟^⊥⊂𝒟^⊥, M is curvature adapted. Since the horosphere H_3 is not of 1-type by Lemma <ref>, M could be geodesic hypersphere with principal curvatures μ =  r,  α = 2  (2r) or a tube about ℍH^m-1 with ν = tanh r,  α = 2  (2r). In either case α = μ + 1/μ,   f = n μ + 3/μ. Taking the metric product of (<ref>)  with σ (Y, ξ)  with X, Y ∈𝒟 we get AX = (- f/2) X, i. e. f + 2 μ = 0. However,

    f + 2 μ = ( n μ + 3/μ) + 2 μ = (n + 2) μ + 3/μ≠ 0,

since μ is positive, contradicting the above.




For hypersurfaces of class A_1 (geodesic hyperspheres and tubes about quaternionic hyperplanes) we have

 

  (i) A geodesic hypersphere in ℍP^m of any radius r ∈ (0, π/2),   r ≠^-1√(3/ (4m + 1)) is of 2-type 
in H(m+1). A geodesic hypersphere in ℍH^m of arbitrary radius r > 0 is of 2-type in H^1(m + 1) via  and the same 
holds for a tube of arbitrary radius r > 0 about a totally geodesic quaternionic  hyperbolic hyperplane 
ℍH^m-1⊂ℍH^m. These statements are also valid for any open portion of the respective submanifolds.

  (ii)   The only  mass-symmetric 2-type hypersurfaces of class A_1 in ℍQ^m(4c)
are open portions of  geodesic hyperspheres of radius r = ^-1√(1/m) in ℍP^m(4).



   
For a geodesic sphere P_1^0(r) in ℍP^m or H_1^0(r) in ℍH^m (class A_1 in ℍP^m and A^'_1 in ℍH^m) 
define

    _c(r) =  r,    when  c = 1  (projective case)
     r,  when  c = - 1  (hyperbolic case)

to be the circular or hyperbolic version of cotangent and denote tan_c(r) = 1/_c(r).
Let μ = _c (r)  be the principal curvature of multiplicity 4(m-1) = n-3 and α = 2 _c (2r) the principal curvature of A |_𝒟^⊥ of multiplicity 3, whereas μ = tanh r,  α = 2  (2r) for a tube about a complex hyperbolic hyperplane 
ℍH^m-1(-4) of class A^''_1. Then

    α = μ - c/μ,    f = n μ - 3c/μ,    f_2 = n μ^2 + 3/μ^2 - 6c,   μ _q = μ ,  ∀ q = 1, 2, 3 .


Setting τ = μ and α_k = α in  (<ref>) we get

    [(n + 2)c - 3μ^-2]  a = c (3n+2) (n + 2) μ^2 + (3n^2 + 2n + 4) - 3c (2n+3)/μ^2 - 9/μ^4.

We may assume that (n+2)c ≠ 3/μ^2 , certainly true when c = -1, and when c = 1 the equality would lead to
μ = √(3/(n+2)), i.e. to r = ^-1√(3/(4m+1)). However, the geodesic hypersphere of this radius in ℍP^m(4)
is of 1-type (see <cit.>). Thus, dividing by (n+2)c - 3μ^-2 we get

    a = (3n + 2) μ^2 + 3/μ^2 + c (3n + 5) = (μ^2 + c) (3n + 2 + 3c/μ^2).

Then from (C_1) or (C_2)we find

    b = 2(n + 1)[ n μ^4 + c (2n+3)μ^2 + 3c/μ^2 + (n + 6)] .

Setting τ = μ,  α_k = α in (C_3) and solving (C_3) for a  gives the same value as in (<ref>), so the conditions (C_1) - (C_3) are consistent and satisfied by the above
values of a and b, the condition (C_4) being trivially satisfied for any class-A hypersurface by (<ref>). According to Lemma <ref>, the equation (<ref>) then. Moreover, the polynomial P(λ ) = λ^2 - a λ + b has two distinct real roots 
λ_u = 2 (n + 1)(μ^2 + c) and λ_v = n μ^2 + 3/μ^2 + c (n + 3) = 1/μ^2(n μ^2 + 3c)(μ^2 + c), 
which are the two eigenvalues of the Laplacian from the 2-type decomposition of . These values are different for if 
λ_u = λ_v then either μ^2 = 1 (c = -1 case) or μ^2 = 3/n+2 (c = 1 case). First condition would lead to 
r = ± 1 which is not possible and the second condition gives a 1-type geodesic hypersphere. By Lemma 1 and a result of <cit.> these are not of 1-type as long as r ≠√(3/(n + 2)) in the projective case and therefore all other geodesic hyperspheres and tubes about ℍH^m - 1 are indeed of 2-type. We can find the actual 2-type spectral decomposition
x = x_0 + x_u + x_v from

    x_u = 1/λ_u (λ_u - λ_v) (Δ^2 x - λ_v Δx) ,     x_v = 1/λ_v (λ_v - λ_u)  (Δ^2 x - λ_u Δx).

Moreover, in this case (<ref>) gives

    Δx = - (n μ - 3c/μ) ξ - 3  (ξ , ξ) - ∑_e_i∈𝒟 (e_i, e_i)

and (<ref>) reduces to

    Δ^2 x   =    -  [n^2 μ^3 + c (3n^2 - 2n - 12)μ - 3(2n-3)/μ - 9c/μ^3] ξ
             +  [(n^2 - 4n - 6) μ^2 - 9/μ^2 - 6cn]  (ξ , ξ) 
    - 2 (n+1)(μ^2 + c ) ∑_e_i ∈𝒟 (e_i, e_i).

Then from the above formulas we get

    x_u    =   1/8m(μ^2 + c)^2 [- 8c(m-1) μ ξ + 4(m-1) μ^2  (ξ , ξ) 
    - (μ^2 + c) ∑_e_i∈𝒟 (e_i, e_i)],
    x_v    =    - μ/(μ^2 + c)^2  [(μ^2 - c) ξ + μ (ξ, ξ )].


Using the expression

    x = 1/m+1 I - c/8(m+1)[  4  (ξ , ξ ) + ∑_e_i∈𝒟σ (e_i, e_i)]

obtained from Lemma 2 of <cit.>,  
we find the center of mass x_0 =  - x_u - x_v to be

    x_0 = I/m+1 + m μ^2 - c/m (μ^2 + c)^2 [μ ξ + 1/2 (ξ, ξ) + (μ^2 + c)( - I/m+1) ].

Since in this equation ξ-component is the only part tangent to ℍQ^m, it follows that the submanifold is mass symmetric
(x_0 = I/(m+1)) if and only if m μ^2 = c. So, c = 1 and mass-symmetric 2-type geodesic hyperspheres exist in ℍP^m(4) only and are 
of radius r = ^-1( 1/√(m)). 
0.3truecm


0.2truecm
Of the two eigenvalues λ_u,   λ_v of the Laplacian for a geodesic sphere given above, in the quaternionic hyperbolic space λ_v is the smaller one, whereas in the projective space which one of the two is smaller and which one is bigger depends on whether the radius of a geodesic sphere is smaller or greter than ^-1√(3/(n + 1)). In any case, one can obtained upper estimates for the first two eigenvalues λ_1,  λ_2 of the Laplacian on a geodesic sphere in ℍQ^m(4 c ). In the projective case we have λ_1 ≤min{λ_u, λ_v } and λ_2 ≤max{λ_u, λ_v }. In the hyperbolic case, λ_1 ≤(n - 3/μ^2)(μ^2 - 1) and λ_2 ≤ 2 (n + 1)(μ^2 - 1) where μ =  r. I the case of a tube about a canonically embedde ℍH^m-1⊂ℍH^m, λ_u, λ_v are negative and λ_u < λ_v

0.3truecm (i) There are no 2-type hypersurfaces in ℍH^m(-4) of class A_2, i.e. no 2-type tubes about canonically 
embedded ℍH^k ⊂ℍH^m,   1 ≤ k ≤ m-2. A hypersurface of class A_2 in ℍP^m(4) is of 2-type if and only 
if it is an open portion of either (a) the tube of radius r = ^-1√(k+1/m-k) or (b) the tube of radius 
r = ^-1√(4k+3/4(m-k)+ 1), about a canonically embedded, totally geodesic  ℍP^k(4) ⊂ℍP^m(4), 
for any k = 1, 2, ..., m-2.
0.3truecm(ii) The only mass-symmetric 2-type hypersurfaces of class A_2 are those in the first series of tubes (a) above.


 (i)Let μ =  r,  ν = - tan r = - 1/μ for model hypersurface of class 
A_2 in ℍP^m
and μ =  r,  ν = tanh r =  1/μ for model hypersurface of class A_2 in ℍH^m.
Then μ,  ν
have respective multiplicities 4l and 4k for some positive integers k,   l with l = m - k - 1 i.e. n = 4l + 4k + 3. Moreover, from Table 1 and (<ref>) we get


    μ ν = - c,   α = μ - c/μ= μ + ν ,   μ_q = μ,   ν_q = ν,   ∀ q = 1, 2, 3.


    f = L μ + K ν,    f^2 = L^2μ^2 + K^2 ν^2 - 2c KL,    f_2 = L μ^2 + K ν^2 - 6 c,

where K:= 4k + 3 and L:= 4l + 3. Our goal is to examine when the three equations (C_1) - (C_3) are consistent and when constants a and b
can be found to satisfy them (As we know, the condition (C_4) is satisfied by every class-A_2 hypersurface ). That comes down to the pair of
equations consisting of (C_3) and (<ref>), having the same solution for a for either value of τ∈{μ, ν}.
Consider the equation (<ref>) in which τ = μ,
multiplied by [2c + f (ν - α )] = (2c - f μ) and the same equation with τ = ν multiplied by (2c - f ν).
Subtract the two multiplied equations to eliminate a. We get

    f   [f_2 + f^2 - c (n + 1) ] + 2 α f   (f + α) -   4  c α = 0.

This is a necessary and sufficient condition for a to have the same value from (<ref>), regardless of the choice of τ∈𝔰(𝒟). 
On the other hand, subtracting the two equations obtained from (<ref>) for τ = μ,  ν, gives

    a f = f   [f_2 + c (3n + 23) ] +   4  c α.

Similarly, from the two equations contained in (C_3) for τ = μ,  ν by subtracting we get

    a = 2 f_2 + f^2 + 2 α (f + α) + 2c (n + 11),

and by eliminating a from these two equations we get exactly the same condition  (<ref>) as before. Moreover, assuming (<ref>), 
we check that (<ref>) and (<ref>) are consistent, so there is only one condition, namely (<ref>), to be satisfied in order 
to make (C_1) - (C_3) consistent,
regardless of the choice of τ , and enable us to solve for a and b. Substituting  the values from (<ref>) and (<ref>) 
into (<ref>), using
α = μ + ν we get

    0   =      L (L + 1)(L+2)μ^3 + K (K+1)(K+2)ν^3
         - c μ  (3 L^2K + 3 L^2 + 6LK + 8L + 2K + 4 )
         - c ν  (3 LK^2 + 3 K^2 + 6LK + 8K + 2L + 4).

With ν = - c/μ this yields

    [(L+1)μ^2 - c (K+1)] [ L(L+2) μ^4 - 2c (LK + K + L + 2) μ^2 + K(K+2)] = 0,

which has the following three solutions

    (a)  μ^2 = (K+1)c/L + 1       (b)  μ^2 = Kc/L + 2      
    (c)  μ^2 = (K+2)c/L.

Clearly, when c = - 1 none of them is possible, so there are no 2-type hypersurfaces of ℍH^m(-4) among class-A_2 hypersurfaces.
When c = 1 from (<ref>) we find

    a = (L^2 + 4L + 2) μ^2 + (K^2 + 4K + 2) ν^2 - 2 LK.

Since L + K = n + 3 = 4m + 2 and with c = 1 , 2 c (n + 3) + α f = (L + K) + L μ^2 + K ν^2, substituting these and a from (<ref>) into (C1) we obtain

    b =         L(L + 1)(L + 2) μ^4 + (L^3 - L^2 K + 2 L^2 + 2 L K + 2 L + 2 K) μ^2
       +  K (K + 1)(K + 2) ν^4 + (K^3 - L K^2 + 2 K^2 + 2 L K + 2 L + 2 K)  ν^2 
        - L^2 K - L K^2 - L^2 - K^2 + 4 L K + 2 L + 2 K.

From these and a = λ_u + λ_v ,    b = λ_u λ_v, we find the two eigenvalues of the Laplacian 
from the 2-type decomposition to be

    λ_u    =    (L+1)(L+2)μ^2 + (K+1)(K+2)ν^2 - (L + K + 2LK), 
    λ_v    =    Lμ^2 + Kν^2 + L + K,      μ =  r,   ν = - tan r .

In the case (a), we get λ_u = 2(n + 5) = 8 (m + 1),  λ_v = 2 (n+3) - (l-k)^2/(l+1)(k+1) = 8(m+1)(LK+ 2m + 1)/(L+1)(K+1), 
λ_u > λ_v, so the
hypersurface is of 2-type. Since μ^2 = ^2r = K+1/L+1, it follows from the Bernd's list that the hypersurface is congruent to an 
open portion of the tube of radius
r = ^-1√(K+1/L+1) = ^-1√(k+1/m-k) about a canonically embedded
ℍP^k(4) ⊂ℍP^m(4), for any  k = 1,..., m-2. In case (b), (<ref>) yields

    λ_u = 8 (m+1)K+1/K = 8 (n + 5) (k + 1)/4 k+3 ,   λ_v = 8 (m+1) L+1/L+2 = 8 (n + 5) (l + 1)/4 l + 5 ,

λ_u > λ_v.
Since μ^2 = ^2r = K/L+2, we identify such hypersurface as an open portion of the tube of radius
r = ^-1√(K/L+2) = ^-1√(4k+3/4(m-k)+1) about a canonically embedded
ℍP^k(4) ⊂ℍP^m(4), for any  k = 1,..., m-2. In ℍP^m(4) cases (b) and (c) generate the same set of examples each being a tube of appropriate radius over one of a pair of alternate focal submanifolds ℍP^k and ℍP^l. Thus, there is no need to consider the last case. 
0.5truecm(ii) For a class-A_2 hypersurface we have from (<ref>) 

    λ_u - λ_v = (L^2 + 2L + 2) μ^2 + (K^2 + 2K + 2) ν^2 -  2 (L+K+  LK).

Note that from Table 1  and the accompanying discussion, in addition to principal curvature α = 2 _c(2r) 
an A_2-hypersurface has also two more principal curvatures μ = _c r and ν = - ctan_c r, 
with corresponding principal subspaces V_μ and V_ν, being J_q-invariant.
Then from (<ref>) and (<ref>) for a split basis of principal directions { e_i} in 𝒟 = = V_μ⊕ V_ν we get

    Δx = - (L μ + K ν) ξ -3  σ (ξ , ξ) - ∑_e_i ∈ V_μ (e_i, e_i) - ∑_e_j ∈ V_ν (e_j, e_j),
 

    Δ^2 x =     - {L^2 μ^3 + K^2 ν^3 + [ L^2 + 4(2m - 1) L - 12] μ + [ K^2 + 4(2 m - 1) K - 12] ν} ξ
       + [(L^2- 4 L -  6) μ^2 + (K^2 -  4 K -  6)ν^2 - 2LK ]  (ξ , ξ)
       - 2 (L+1)(μ^2 + 1) ∑_e_i ∈ V_μσ (e_i, e_i) - 2 (K+1)(ν^2 + 1)∑_e_j ∈ V_ν (e_j, e_j).

Then x_u and x_v can be computed as in (<ref>). Since we now assume the hypersurface of ℍP^m to be of type 2 and mass-symmetric 
via x we must have x_0 = x - (x_u + x_v) = I/(m+1). Because I and x are normal to x (ℍP^m), a necessary condition 
for mass-symmetry in H^(1)(m+1) is that the ξ-component of x_u + x_v be zero. 
0.2truecm
Observing the corresponding values of λ_u ,  λ_v in each of the cases we see that the ξ-component of x_u + x_v
for hypersurfaces in (b) is never zero, whereas for hypersurfaces of case (a) this component is identically equal to zero. 
Namely, since a = λ_u + λ_v and b = λ_u λ_v from (<ref>) we have

    x_u + x_v = 1/λ_u λ_v[ (λ_u + λ_v) Δx - Δ^2 x] = 1/b  (a Δx - Δ^2 x).

From this and the preceding formulas, the ξ-component of - (x_u + x_v) of any hypersurface belonging to case (b) when multiplied by ν is found to be equal to 64 (m + 1)/b K ≠ 0, whereas using the values μ^2 = K + 1/L + 1,  ν^2 = L + 1/K + 1  to reduce powers μ^3 and ν^3 to multiples of μ and ν, K + L = 4m + 2, and (<ref>), (<ref>), (<ref>), refering to (<ref>) we compute the ξ-component of (<ref>) for any hypersurface in (a)  to be 

    2/b{(L + 1)[LK + 4 (m - 1)] μ  + (K + 1) [LK + 4 (m - 1)] ν} ,
which yields 0 when multiplied through by μ.
By (<ref>) we know that

    I/m + 1 - x = 1/8(m + 1)[4 σ (ξ , ξ) + ∑_e_i ∈ V_μσ (e_i, e_i) + ∑_e_j ∈ V_νσ (e_j, e_j)  ].

Then, further straightforward computations show that σ (ξ , ξ)-component and ∑σ(e_i, e_i) components of - (x_u + x_v) where  e_i belongs to V_μ or V_ν  are equal to corresponding components of the right-hand side of (<ref>), so all the components
on both sides of a mass-symmetric 2-type decomposition of x are matched and with this, part (ii) is also proved. 

0.3truecm
The two families of tubes referred to in Lemma 4 have also another representation.
Let

    M_4k+3, 4l+ 3(r):= S^4k+3(cos r) × S^4 l+3(sin r),   0 < r < π / 2,

be the family of generalized Clifford tori in an
odd-dimensional sphere S^n+4⊂ℍ^m+1,   n = 4m-1. By choosing the two spheres (with the indicated radii) in the above
product to lie in quaternionic subspaces we get the fibration S^3 → M_4k+3, 4 l+3(r) → M^ℍ_k, l(r):= π (M_4k+3, 4 l+3(r)) 
compatible with the Hopf fibration π : S^n+4→ℍP^m(4), which submerses M_4k+3, 4 l+3(r) onto M^ℍ_k, l(r).
Pérez and Santos show <cit.> that M^ℍ_k, l(r) is a tube of radius r about totally geodesic ℍP^k(4)
with principal curvatures r,   - tan r,  2  (2r) of respective multiplicities 4 l, 4 k, and 3. Accordingly, the family
of hypersurfaces corresponding to the case (a) is given as open portions of
0.2truecm
M^ H_k, l(r) = π(S^K( √(K+1/n+3))×
S^L( √(L+1/n+3))  ),  ^2r = K+1/L+1,
and the family of hypersurfaces corresponding to the case (b) is
0.2truecm
M^ H_k, l(r) = π(S^K( √(K/n+3))×
S^L( √(L+2/n+3))  ),   ^2r = K/L+2,
where for both families K = 4 k+ 3 and L = 4 l+ 3 are odd positive integers with K + L = n + 3 = 4 m + 2. The family of hypersurfaces corresponding to the case (c) is the same family as 
in (b), with the roles of K and L interchanged and the spherical factors in the above representation reversed. Hypersurfaces of case (c) can be also described as tubes over 
ℍP^k(4) of radius r = ^-1√(4k+5/4(m-k)-1), for k = 1, 2, … , m-2, but are not examined as a separate
case since they constitute the same family as the one under case (b). Namely, the tube about ℍP^k(4) of this radius r is the 
same as the tube over the other focal variety ℍP^l(4) of radius π/2 - r = ^-1√(4 l+3/4(m-l)+1), 
which appears within family (b).
0.4truecmRemark. Note that according to a result of Barbosa et al. <cit.>, the tube of radius r over ℍP^k(4)  in ℍP^m(4)   is stable with respect to normal
variations preserving the enclosed volume if and only if 
^-1√(4 k+5/4(m-k)-1)≤  r ≤^-1√(4 k+ 3/4(m-k)+1)  . Hence, the 2-type tubes over ℍP^k(4) of radii ^-1√(4 k+ 3/4(m-k)+1)
and ^-1√(4k+5/4(m-k)-1)
are distinguished by being the largest, respectively the smallest, stable tubes about ℍP^k, for each  k = 1, 2, ..., m-2,
i.e. the values of radii in cases (b) and (c) are precisely the endpoints of the stability interval for r.
0.4truecm

There exist no class-B hypersurfaces of ℍ H^m(- 4) which are of 2-type via x, i.e. no 2-type hypersurface H_2(r) in ℍH^m(- 4) as given in Table 1. A class-B hypersurface of ℍ P^m(4) is of 2-type via x if and only if it is an open portion of one of the following


  (i)   The tube of radius r = 1/2^-1(1/√(m))  about a canonically embedded, totally geodesic  ℂP^m(4) ⊂ℍP^m(4); 


  (ii)   The tube of radius r = 1/2^-1√((3 + √(96 m^2 - 15))/2 (4m^2 - 1)) about a canonically embedded, totally geodesic  ℂP^m(4) ⊂ℍP^m(4). 

0.2truecm
In both of these cases, the hypersurface is also mass-symmetric.



For class-B hypersurfaces and the values of their principal curvatures given in Table 1, observing the notation introduced in Lemma 3 we have the principal curvatures of A|_𝒟

    μ = _c (r),  ν = - c tan_c (r),   both of multiplicity   2 (m - 1)

and those of A|_𝒟^⊥,

    α := α_1 = 2 _c( 2 r),   α_2 = α_3 = - 2 c tan_c (2 r),

where α has multiplicity 1 and the other one multiplicity 2. Then

    μν = - c,   μ + ν = α,   α_2 = α_3 = - 4 c/α


    ∑_q = 1^3 α_q = α - 8 c/α,     ∑_q = 1^3 α_q^2 = α^2 + 32/α^2


    f = (2 m - 1) α - 8c/α,      f^2 = (2m - 1)^2 α^2 + 64/α^2 -  16 c (2 m - 1)


    f_2 = (2m - 1) α^2 + 32/α^2 + 4 c (m - 1) .  4truecm

For every τ∈𝔰 (𝒟) we find from (<ref>) that τ_q = (2 c + α_q τ)/(2 τ - α_q), from which we get

    μ_1 = μ ,  μ_2 = μ_3 = ν  and  ν_1 = ν,   ν_2 = ν_3 = μ

and hence for both τ = μ and τ = ν we have

    τ + ∑_q τ_q = 2 α,     τ^2 + ∑_q τ_q ^2 = 2 α^2 + 4 c   .


In (C_1) substitute α_1 and  α_2 for α_k and subtract the two formulas to get

    f a    = f  [f_2 + c (3n + 7)] - 4 c∑_q α_q 
       = (2m - 1)^2 α^3 + 4 c (8 m^2 - 8 m + 1) α - 64 m/α - 256 c/α^3.

and then substututing this back into (C_1) we get

    (n + 3) a = b/2  c + 2 α^2 + 128/α^2 + 8 c (4 m^2 + 1).

In (C_2) cancell out τ fa = τ{f  [f_2 + c (3 n + 7)] - 4 c ∑_q α_q } on both sides, use (<ref>) - (<ref>) and n = 4m - 1 to get

    (n + 4) a = b/2  c + (4 m^2 + 4 m + 1) α^2 + 64/α^2 + 4 c (8 m^2 + 4 m + 1)

Subtracting (<ref>) from (<ref>) gives

    a = (4 m^2 + 4 m - 1) α^2 - 64/α^2 + 4 c (4 m - 1)

and substituting this in either (<ref>) or (<ref>) yields

    b = 8 c (m + 1) [(4 m^2 + 2 m - 1) α^2 - 64/α^2 + 4 c (2m - 1)  ]

Joint validity of the formulas (<ref>) - (<ref>) makes (C_1) and (C_2) consistent. Next, cancel out f a from both sides of (C_3) using (<ref>) and use (<ref>) - (<ref>) on the right hand side to simplify and solve for a to get exactly the same value as in (<ref>), which means that (C_3) is consistent with the previous formulas. Finally, to make (<ref>) consistent with (<ref>), multiply the latter by f from (<ref>) to obtain 

    f a = (2m - 1)(4 m^2 + 4m - 1) α^3 - 4 c (14m - 3) α - 32(8m- 3)/α + 512 c/α^3 ,

which must be the same as (<ref>) for suitable values of α^2. Equating fa from these two equations results in the equation

    m (4 m^2 - 1) α^6 - 4 c (4m^2 + 3m - 1) α^4 - 48 (2m - 1) α^2 + 384 c = 0 .

The left-hand side factors as

    (m α^2 - 4c)[(4m^2 - 1) α^4 - 12 c α^2 - 96 ],

from where we get the solutions of (<ref>) to be

    α^2 = 4 c/m,   α^2 = 6 c ± 2 √(96 m^2 - 15)/4m^2 - 1.

Since in the hyperbolic case c = - 1 and α^2 = 4 ^2(2 r) > 4, we see that none of these works in that case, so there are no hypersurfaces of class B in ℍH^m(- 4) which are of 2-type. In the projective case  (c = 1) we get two solutions for α = 22r and the corresponding values for the radii

    r = 1/2 arccot(1/√(m))  ,    r = 1/2 arccot(√(3 + √(96 m^2 - 15)/2(4 m^2 - 1)) ) ,
 
which both satisfy the conditions (C_1) - (C_3) of Lemma 1. There is also the condition (C_4) which is not automatically satisfied since the relation (<ref>) does not hold for class-B hypersurfaces. However, we will show  by direct calculation that the tubes of these two radii about canonically embeded ℂP^m(4) ⊂ℍH^m(4) are mass-symmetric and of 2-type by verifying that the mass-symmetric 2-type equation 

    Δ^2 x - a Δx + b (x - I/m+1) = 0

holds for these hypersurfaces. Namely, assume that the basis of quaternionic structure { J_q }, q = 1, 2, 3 has been chosen locally so that U_q = - J_q ξ is a principal direction for α_q and {e_i  } is a basis of principal directions. Using (<ref>) - (<ref>), formula (<ref>) gives

    Δx = - [(2m - 1)α - 8c/α] ξ - 3 σ (ξ, ξ) - ∑_e_i ∈𝒟σ (e_i, e_i)

and (<ref>) reduces to

    Δ^2 x =    - f a  ξ + {(4 m^2 - 4 m - 1) α^2 - 64/α^2 - 4 c (4m + 1)  } σ (ξ, ξ) 
        - 2 c (4m + 3) ∑_e_i ∈𝒟σ (e_i, e_i ) - 2 f ∑_e_i ∈𝒟σ (e_i, Ae_i ) -  2 ∑_e_i ∈𝒟σ (Ae_i, Ae_i ) ,

where fa is given by (<ref>). Let the notation {·}_mod Ξ mean taking only terms of the expression {·} that involve ξ and σ (ξ, ξ) and {·}_mod Ξ^⊥
taking those terms that involve σ (e_i, e_i).
Since the multiplicity of both μ and ν is 2(m - 1), from the discussion about the formula (<ref>) and using (<ref>) and (<ref>) it follows that there is an orthonormal basis of principal directions of 𝒟,

    {e_i }∪{J_1 e_i }∪{J_2 e_i  }∪{J_3 e_i },    i = 1, 2, … , m-1

so that {e_i }∪{J_1 e_i } spans V_μ and {J_2 e_i  }∪{J_3 e_i } spans V_ν,   i = 1, 2, … , m-1.
Then, using this basis for 𝒟 and observing (<ref>), (<ref>) we compute

    {Δ^2 x}_mod Ξ^⊥   =  -  2  { (4m + 3) c∑_τ = μ , ν
     e_i ∈ V_τσ(e_i, e_i) + ∑_τ = μ , ν
     e_i ∈ V_τ  (f τ + τ^2) σ (e_i, e_i) }
       = -  2  { (4m + 3) c ∑_i = 1^m - 1  4  σ(e_i, e_i) +  (f μ + μ^2)∑_i = 1^m - 1   2 σ (e_i, e_i) +  (f ν + ν^2)∑_i = 1^m - 1   2 σ (e_i, e_i)    }
       = - 2 {4c (4m + 3) + 2   f (μ + ν) + 2 (μ^2 + ν^2)  }∑_i = 1^m - 1σ (e_i, e_i)
       = - 8m (α^2 + 4c)  ∑_i = 1^m - 1σ (e_i, e_i) = - 2m (α^2 + 4c)  ∑_e_j ∈𝒟σ (e_j, e_j)   ,

and, likewise, from (<ref>) and (<ref>)

    {Δx}_mod Ξ^⊥ = -   ∑_e_i ∈𝒟σ (e_i, e_i),     {x - I/m + 1}_mod Ξ^⊥ = - c/8 (m + 1)∑_e_i ∈𝒟σ (e_i, e_i).

Therefore,  when c = 1, with values of a and b from (<ref>) and (<ref>) it follows that mod Ξ^⊥- component of the left-hand side of (<ref>) is zero and in a similar way, using (<ref>), (<ref>),   mod Ξ  - component is also zero, which means that equation (<ref>) is satisfied, showing that the immersion x is mass-symmetric and of 2-type. For the hypersurface M given in (i) we have 

    λ_u + λ_v = a = 4 (m + 1)(4m - 1)/m ,     λ_u λ_v = b = 32 (m + 1)^2(2m - 1)/m,

hence λ_u = 4 (m + 1)(2m - 1)/ m and λ_v = 8 (m + 1) belong to the spectrum of M. Moreover, using

    x_u = 1/λ_u - λ_v[ Δx - λ_v (x - I/m + 1) ],     x_v = 1/λ_v - λ_u[ Δx - λ_u (x - I/m + 1) ]

and (<ref>) we find an explicit spectral resolution of immersion x into vector eigenfunctions:

    x_u    = - √(m)/2(m + 1) ξ - m/4 (m + 1) σ (ξ , ξ)
    x_v    = √(m)/2 (m + 1) ξ + m - 2/4 (m + 1) σ (ξ , ξ) - 1/8(m + 1)∑_e_i ∈𝒟σ (e_i, e_i).

plus the center of mass x_0 = I/(m + 1).

One example of a  mass-symmetric 2-type hypersurface of class B is obtained when we take m = 3 to get an 11-dimensional real hypersurface of ℍP^3 which is the tube of radius r = π/6 about a canonically embedded ℂP^3. When m = 4, the second value of r gives an example of 15-dimensional hypersurface of 2-type in ℍP^4, which is the tube of radius π/6 about ℂP^4.
Except for the two tubes given in Lemma 5, all other hypersurfaces of class B in ℍP^m are of 3-type.
0.3truecm

A tube of any radius r > 0 about canonically embedded ℂH^m(- 4) in ℍH^m(4) is mass-symmetric and of 3-type. The same is true for a tube of any radius r ∈ (0, π/4) about canonically embedded ℂP^m(4) in ℍP^m(4), except for the two tubes listed in Lemma 5 which are of 2-type.



The proof goes by direct computation of iterated Laplacians and matching corresponding componnets of 3-type mass-symmetric equation. We begin by computing the Laplacian of σ(ξ, ξ). Since

    ∇_X σ(ξ, ξ) = - 2 c   [X + ∑_q = 1^3 ⟨ U_q, X⟩ U_q ] -   2 σ (AX, ξ),

by repeated use (<ref>) and (<ref>) we further compute

    Δ (σ (ξ, ξ)) =    ∑_i = 1^n [ ∇_∇_e_ie_iσ (ξ , ξ) - ∇_e_i∇_e_iσ (ξ , ξ)  ]  
    
    =      2 c [ f  + ∑_q α_q ] ξ + 2 c ∑_q = 1^3 [ ∇_U_q U_q + ∑_i = 1^n ⟨∇_e_i U_q,   e_i⟩  U_q ]
        -   2  ∑_i = 1^n  A_σ (A e_i , ξ) e_i + 2 (3 c + f_2) σ (ξ , ξ)
          +  2  ∑_i = 1^n σ ((∇_e_iA)e_i ,  ξ) + 2 c  ∑_i = 1^n σ (e_i, e_i) - 2 ∑_i = 1^n σ (A e_i, A e_i).

Then employing (<ref>) and (<ref>) we find

    ∑_q = 1^3 [ ∇_U_q U_q + (div  U_q)  U_q ] = 0,     ∑_i = 1^n (∇_e_iA) e_i = tr (∇ A) = ∇ f = 0

and using (<ref>) and (<ref>) - (<ref>) arrive at

    Δ (σ (ξ , ξ)) = 4c  (α - 8 c/α) ξ + 4  [(2m + 1) c + (m - 1)α^2] σ (ξ , ξ) - α^2  ∑_e_i ∈𝒟σ (e_i, e_i).

For a curvature-adapted hypersurface we have ∑_q  S_q A U_q = 0. Using (<ref>) - (<ref>) combined with (<ref>), formula (<ref>) yields

    ∑_e_i ∈𝒟Δ (σ (e_i , e_i)) =      8 c (m - 1)[(2m + 3) α - 8 c/α]  ξ - 8 (m - 1)(c + 2 α^2)  σ (ξ , ξ) 
          + 4 [ 2 c (m + 1) + α^2 ]  ∑ _e_i ∈𝒟σ (e_i,  e_i) .

Further, the expressions (<ref>), (<ref>) and (<ref>) reduce, respectively, to (<ref>) and


    Δξ =    [ (2m - 1) α^2 + 32/α^2 + 8 c (m -1)  ]  ξ  - [(2m - 3) α + 8 c/α]  σ (ξ , ξ) + α ∑_e_i ∈𝒟σ (e_i,  e_i),



    Δ^2 x =      - [ (2m - 1)^2 α^3 + 4 c (8 m^2 - 8 m + 1) α - 64 m/α - 256 c/α^3]  ξ
         + [(4 m^2 - 4 m - 1) α^2 - 64/α^2 - 4 c (4 m + 1)  ]  σ (ξ , ξ) 
         - 2 m (α^2 + 4 c) ∑_e_i ∈𝒟σ (e_i,   e_i).

Taking the Laplacian of (<ref>) and using (<ref>) - (<ref>) we have


    Δ^3 x =      - {(2m - 1)^3 α^5 + 8 c  (16 m^3 - 20 m^2 + 6m  - 1) α^3    
              + 16   (24 m^3 - 28 m^2 + 6 m  - 1) α - 512 c (2m - 1)/α - 4096 m/α^3 - 8192 c/α^5} ξ
          + { (24 m^3 - 20 m^2 - 6m + 1) α^4 + 8 c   (12 m^3 - 8 m^2 - 6 m + 1) α^2 
              - 512 c (3m - 1)/α^2 - 2048/α^4 + 16  (4 m^2 - 30 m + 17)} σ (ξ , ξ) 
         - { 8 m^2 α^4 + 48 c m^2 α^2 - 256 c/α^2 + 64 (m^2 - 1) } ∑_e_i ∈𝒟σ (e_i , e_i).

It is now a direct verification that x satisfies the equation for a mass-symmetric 3-type hypersurface, viz.

    Δ^3 x + p Δ^2 x + q Δx + r (x - I/m + 1) = 0

where 

    p = - 1/α^2 (α^2 + 4 c) [(6 m - 1) α^2 + 8 c] ,    r = - 16 c/α^2  (2 m^2 + m - 1)(α^2 + 4 c)^2 (m α^2 + 4 c)
.

    q = 1/α^2 (α^2 + 4 c) [4m (2m - 1) α^4 + 8 c (6 m^2 + 2 m - 1) α^2 + 32 (4m + 1)].

Given that there are no 1-type hypersurfaces of this kind, when we eclude those two 2-type examples from Lemma 5, we are left with all the other such hypersurfaces which must be of type 3.

Combining together the information from Lemmas 2-5 we obtain

Let M^4m - 1 be a curvature adapted real hypersurface of ℍP^m(4),  m ≥ 2. Then  M is of 2-type in H (m + 1) via Φ if and only if M is locally congruent to an open portion of one of the following
  

  (i)   A geodesic hypersphere of ℍP^m(4) of any radius r ∈ (0, π/2),   r ≠^-1√(3/ (4m + 1));


  (ii)    The tube of radius r = ^-1√(k+1/m-k) about a canonically embedded, totally geodesic  ℍP^k(4) ⊂ℍP^m(4), 
for any k = 1, 2, ..., m-2;


  (iii)   The tube of radius r = ^-1√(4k+3/4(m-k)+ 1) about a canonically embedded, totally geodesic  ℍP^k(4) ⊂ℍP^m(4), 
for any k = 1, 2, ..., m-2;


  (iv)   The tube of radius r = 1/2^-1(1/√(m))  about a canonically embedded, totally geodesic  ℂP^m(4) ⊂ℍP^m(4); 


  (v)   The tube of radius r = 1/2^-1√((3 + √(96 m^2 - 15))/ 2 (4m^2 - 1)) about a canonically embedded, totally geodesic  ℂP^m(4) ⊂ℍP^m(4). 




As proved in <cit.>, a curvature-adapted hypersurface of ℍP^m(4) has constant principal curvatures and is to be found in the left portion of Table 1. Then, the Lemmas 3-5 finish the proof.

Likewise,


Let M^4m - 1 be a curvature adapted real hypersurface of ℍH^m(- 4) with constant principal curvatures. Then  M is of 2-type in H^1 (m + 1) via Φ if and only if M is locally congruent to an open portion of one of the following
  

  (i)   A geodesic hypersphere of ℍH^m (- 4) of any radius r > 0;


  (ii)    The tube of arbitrary radius r > 0 about a totally geodesic quaternionic  hyperbolic hyperplane ℍH^m-1(- 4) of ℍH^m(- 4). 





Among 2-type hypersurfaces of ℍQ^m(4 c) listed in Theorems 2 and 3

  

  (i)   The only ones that are of 2-type and mass-symmetric in an appropriate hyperquadric of 𝐇^(1) (m + 1) centered at I/(m + 1) that contains them are open portions of:   (1)  geodesic hyperspheres of ℍP^m (4) of radius r = ^- 1(√(1/ m)) (2)  the tubes of radius r = ^-1√(k+1/m-k) about a canonically embedded, totally geodesic  ℍP^k(4) ⊂ℍP^m(4), 
for any k = 1, 2, ..., m-2 and (3)  the tubes of radii r = 1/2^-1(1/√(m)) and r = 1/2^-1√((3 + √(96 m^2 - 15))/ 2 (4m^2 - 1)) about a canonically embedded, totally geodesic  ℂP^m(4) ⊂ℍP^m(4). 


  (ii)    The 2-type hypersurfaces that are minimal in ℍQ^m(4c) exist only in the quaternionic projective space and are open portions of geodesic spheres of radius r = ^-1(√(3/(4 m - 1))) and tubes of radius r = π/4 around canonically  embedded ℍP^k (4) in ℍP^2 k + 1(4) when the quaternionic dimension m = 2k + 1 is odd.




The proof of part (i) is contained in Lemmas 3-5. To prove part (ii) we explot the information given in Table 1. Regarding class-A_1 hypersurfaces when k = 0 (geodesic spheres) we have μ = _c r,   m(μ) = 4 (m - 1) and α = 2 _c (2r),   m(α) = 3 as principal curvatures, hence 

    f = tr  A = 4 (m - 1) _c r + 6 _c(2r) = (4 m - 1)_c r - 3c tan_c r,

due to 2 _c(2 r) = _c r - c tan_c r.
So, tr  A can be zero only when c = 1 and ^2 r= 3/(4m - 1) = 3/n, yielding the geodesic sphere in ℍP^m(4) of radius r = ^-1√(3/(4m - 1)) which is indeed of 2-type and minimal. If k = m - 1 in the projective case P_1^m - 1(r) = P_1^0 (π/2 - r) is still a geodesic sphere, yielding the same value of the radius as above. In the hyperbolic space the principal curvatures are ν = tanh r and α = 2  (2 r) in which case tr  A = (4 m - 1) tanh r + 3  r cannot be zero, so no minimal hypersurface of this kind. Searching for minimal 2-type examples among class-A_2 hypersurfaces  we see that formula (<ref>) implies that α = μ + ν = μ - c/μ = 0, i. e. μ^2 = c. So when c =  - 1 we cannot have a minimal 2-type hypersurface in ℍH^m of class A_2. In the projective case it follows that μ =  r = 1 and ν = - 1 so r = π/4 and f = Lμ + K ν = L - K = 0, implying that l = k = m - 1/2
so m = 2 k + 1 is necessarily odd, resulting in class-A_2 hypersurface M^ℍ_k, k (π/4) = π(S^4 k + 3(1/2) × S^4k + 3(1/2) ), which is indeed minimal and of 2-type  as first observed by Chen <cit.>, using a different method. It is the tube of radius π/4 about a canonically embedded totally geodesic ℍP^k (4) ⊂ℍP^2k + 1(4). Our argument shows,  that it is in fact the only 2-type minimal hypersurface of ℍQ^m(4c) among complete class-A_2 hypersurfaces. So, minimal 2-type hypersurfaces of class A_2 exist only in the projective space and only when m = 2k + 1 is odd and they are congruent to (an open portion of) the hypersurface  M^ℍ_k, k(π/4) described above, which also happens to be mass-symmetric according to Lemma 4.

Curvature-adapted assumption on a hypersurface is seemingly a strong condition, at least in the ℍP^m case, since it implies constant principal curvatures. On the other hand, one does not need this, or any other assumption if the number of principal curvatures of a hypersurface of ℍQ^m is restricted to be at most two. Namely, when m ≥ 3, Martinez and Pérez showed in ℍP^m case and Ortega and Pérez in ℍH^m,  that a hypersurface with at most two distinct principal curvatures at each point  must be an open subset of one of the following:
 (1)  A geodesic hypersphere (2)   A tube over totally geodesic ℍH^m - 1(- 4)
 in ℍH^m(- 4) (3)  A horosphere H_3 in ℍH^m (- 4).
Hence,
0.3truecmCorollary 1. If M is a real hypersurface of ℍQ^m(4c),   m ≥ 3, with at most two distinct principal curvatures at each point and M is not a horosphere, then M is of Chen type 1 when it is (an open portion of) a geodesic hypersphere of radius
 ^-1(3/√(4m+ 1)) in ℍP^m(4) or, otherwise, of Chen type 2.


10

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0.2truecm
1.4truecm Ivko Dimitrić
0.1truecm
1truecm Pennsylvania State University Fayette
0.1truecm
1truecm 2201 University Drive,
0.1truecm
1truecm Lemont Furnace, PA 15456, USA.
0.1truecm
1truecm Email:   ivko"40psu.edu




 Paper Title: Persistent de Rham-Hodge Laplacians in the Eulerian representation.






 
 
 


Recently, topological data analysis (TDA) has become a trending topic in data science and engineering. However, the key technique of TDA, i.e., persistent homology, is defined on point cloud data, which restricts its scope. In this work, we propose persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL) for abbreviation, for the TDA on manifolds with boundaries, or volumetric data.         
Specifically, we extended the evolutionary de Rham-Hodge theory from the Lagrangian formulation to the Eulerian formulation via structure-persevering Cartesian grids, and extended the persistent  Laplacian on point clouds to persistent (de Rham-)Hodge Laplacian on nested families of manifolds with appropriate boundary conditions. 
The proposed PHL facilitates the machine learning and deep learning prediction of volumetric data. For a proof-of-principle application of the proposed PHL, we propose a persistent Hodge Laplacian learning (PHLL) algorithm for data on manifolds or volumetric data. To this end, we showcase the PHLL prediction of protein-ligand binding affinities in two benchmark datasets.  Our numerical experiments highlight the power and promise of PHLL. 








Persistent de Rham-Hodge Laplacians in the Eulerian representation
    Zhe Su^1, Yiying Tong^2,
[Corresponding author.		Email: ytong@msu.edu]
and Guo-Wei Wei^1,3,4 
[Corresponding author.		Email: weig@msu.edu]


^1Department of Mathematics,
 Michigan State University, East Lansing, MI 48824, USA

^2Department of Computer Science and Engineering,
 Michigan State University,
East Lansing, MI 48824, USA

^3Department of Biochemistry and Molecular Biology,
 
Michigan State University, East Lansing, MI 48824, USA

^4Department of Electrical and Computer Engineering, 

Michigan State University, East Lansing, MI 48824, USA
    Accepted for publication in Astrophysics & Space Sciences (ApSS)
=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





§ INTRODUCTION


Recent years have witnessed a fast growth of topological data analysis (TDA) in data science and engineering <cit.>. The growth is driven by the great promise of topological approaches to real-world data that are distinguished from any other statistical, mathematical, physical, and engineering methods. Typically, TDA offers a multiscale topological characterization of data, which is the case with persistent homology <cit.>, a key method employed in TDA. A major feature of persistent homology is its multiscale analysis, which creates a family of topological spaces from the original data to track the topological persistence, i.e., the lifespan of topological invariants across scales. The other major feature of persistent homology is its topological description of a space (like connected components, loops, and voids) in terms of topological invariants, such as Betti numbers.  As such, persistent homology-based TDA  leads to much topological simplification of the geometric information in the data. Consequently, TDA typically works extremely well for data with intricate complexity. Unfortunately, for data without geometric complex, TDA may give rise to an oversimplification or neglect key geometric characteristics, leading to a less competitive approach. 

For many years, persistent homology has been used in qualitative analysis, which is somewhat counterintuitive and unproductive for non-experts. The power of persistent homology was not demonstrated until it was utilized in quantitative and predictive analysis via machine learning algorithms <cit.>. 
Topological deep learning (TDL) was introduced to deal with large and intrinsically complex datasets <cit.>. 
When TDL was coined in 2017, it referred to the topological featurization in deep neural networks <cit.>. More recently, simplicial neural networks and other topological neural techniques have been applied to the design of neural network architecture, making TDL an emerging paradigm in data science and machine learning <cit.>. However, an increasing concern associated with this rising popularity is whether TDL brings any practical benefit beyond its mathematical elegance. There are many applications where TDL has demonstrated superiority to other competitive methods. Perhaps some of the most compelling examples are TDL's dominant wining of D3R Grand Challenges, an annual worldwide competition series in computer-aided drug design <cit.>, its discovery of the mechanisms of SARS-CoV-2 evolution <cit.>, and its successful forecast of emerging dominant SARS-CoV-2 variants BA.2 <cit.> and BA.4/BA.5 about two months in advance <cit.>.     

 It is interesting to understand why TDL (or TDA) was so successful in the aforementioned examples, but was not competitive in many other situations. First, biomolecular data, which are intricately complex in their internal structures, were involved in the above compelling examples. As such, topological simplification was a productive process, whereas TDL leads to the severe loss of crucial information in many other data that are relatively simple in their internal structures. Additionally, it was element-specific persistent homology, rather than persistent homology, that was applied in the above examples. This approach captures physical and biological interactions in the biomolecular data <cit.>.   
  In fact, in the forecast of emerging dominant SARS-CoV-2 variants BA.4/BA.5, persistent Laplacian, rather than persistent homology,  was utilized.  This happens because persistent homology has many drawbacks or limitations <cit.>. 
 First, the topological invariant extracted from persistent homology is qualitative, rather than quantitative. For example, persistent homology does not distinguish a five-number from a six-number ring.  
 Additionally, persistent homology is incapable of dealing with different elements in a point cloud, which is ineffective with the physics and chemistry of (bio)molecular data. 
 Moreover, persistent homology cannot describe non-topological changes, i.e., homotopic shape evolution during the multiscale analysis.
Further, persistent homology is incapable of handling directed networks and digraphs, such as polarization, regulation, and control issues in applications. Finally, persistent homology is unable to characterize structured data, e.g., hypergraphs, directed networks, etc. 
These challenges call for innovative topological methods. 
 
To address these challenges, two types of persistent topological Laplacians (PTLs) were introduced in 2019 <cit.>. The first type of PTL is called evolutionary de Rham-Hodge method for data on manifolds <cit.>. This approach creates a family of multiscale manifolds with boundaries from a given data and then builds Hodge Laplacian operators on the multiscale manifolds with appropriate boundary conditions. The second type of PTL is called persistent spectral graph, also known as persistent combinatorial Laplacian or persistent Laplacian defined for point cloud data <cit.>.  While originated from sharply different topological spaces <cit.>, these two PTLs share the same algebraic structure.  The harmonic spectra of PTLs fully recover the topological invariants of persistent homology. However, the nonharmonic spectra of PTLs capture the homotopic shape evolution during the multiscale analysis that cannot be done with persistent homology. Computational algorithms and mathematical analysis of persistent Laplacian have been reported <cit.>. Persistent Laplacian has been shown to outperform persistent homology in many applications <cit.>.  

In the past few years, much effort has been given to extend persistent Laplacian to further address other limitations of persistent homology on point cloud data, leading to   
persistent sheaf Laplacians<cit.>,
persistent path Laplacians, 
persistent hypergraph and hyperdigraph Laplacians<cit.>,

persistent directed flag Laplacians,

persistent Mayer Laplacians,

and persistent interaction Laplacians
<cit.>.


In addition to point cloud data and data on manifolds, there is another widely available data form, namely, curves embedded in the three-dimensional Euclidean space, such as knots, links, and tangles and their generalizations <cit.>. Evolutionary Khovanov homology has been proposed to deal with embedded curve data <cit.>. It integrates algebraic topology, geometric topology, and metric analysis for the first time. However, effective computational algorithms are to be developed for this approach to be widely used in practical applications.  
Similarly, while case studies have been shown for evolutionary de Rham-Hodge theory for data on manifolds <cit.>, the early approach based on discrete exterior calculus (DEC) <cit.> in the Lagrangian representation is not efficient for multiscale analysis and machine learning studies. Specifically, the regeneration of the evolving manifold at different scales with different simplicial meshes causes numerical inconsistencies and becomes expensive for practical applications in machine learning studies.    

The objective of this work is to develop a persistent de Rham-Hodge theory on the Euler representation. As such, it solves Hodge Laplacians on a pre-designed structure-persevering Cartesian grid for all scales to avoid numerical inconsistency. It constructs a natural mapping of differential forms from a manifold with boundary embedded in ^3 to a large manifold,  uses it to produce persistent cohomology mapping, and constructs a persistent Hodge Laplacian with built-in boundary conditions, extending the persistent combinatorial Laplacian.
Our new approach draws on differential geometry,  algebraic topology, partial differential equation (PDE), metric analysis, and numerical analysis. To give a proof-of-principle demonstration, we pair the proposed persistent de Rham-Hodge Laplacians with an effective machine learning algorithm to predict protein-ligand binding affinities. Based on two benchmark datasets, PDBbind v2007 and  PDBbind v2016, we show that our new approach gives rise to cutting-edge performance.    
The rest of this paper is organized as follows. Sec. 2 offers a primer on the de Rham-Hodge theory on manifolds with boundaries; Sec. 3  presents our discretization for evolutionary de Rham-Hodge theory based on spectrum calculation of Laplacians associated with sublevel sets on Cartesian grids; Sec. 4 presents our construction for persistent de Rham-Hodge Laplacians both in the continuous setting and for given level set functions on Cartesian grids; Sec. 5 showcases preliminary studies on applications; and Sec. 6 concludes the paper.




§ DE RHAM-HODGE THEORY


Let M be an m-dimensional smooth, orientable, compact Riemannian manifold with boundary. Denote by Ω^k(M) the space of all differential k-forms on M, i.e., the space of all smooth antisymmetric covariant tensor fields on M of degree k. The differential d, also called exterior derivative, is the unique -linear mapping from the space of k-forms Ω^k(M) to the space of (k+1)-forms Ω^k+1(M) satisfying the Leibniz rule with respect to the wedge product ∧ and the nilpotent property dd = 0.
A key property of differential forms is that they can be integrated over any orientable k-submanifolds of M. For any oriented (k+1)-submanifold S⊂ M with boundary ∂ S, Stokes' theorem, as a generalization of the Newton-Leibniz rule, states that the integral of a differential k-form ω over ∂ S is equal to the integral of its differential over S, i.e., 

    ∫_S dω= ∫_∂ Sω.

The differential d generalizes and unifies the classical operators in vector calculus, such as gradient ∇, curl ∇×, and divergence ∇· in ^2 and ^3. For instance, in ^3, 0-forms and 3-forms can be identified with scalar fields, while 1-forms and 2-forms can be identified with vector fields. In this case, the differential d corresponds to the gradient operator ∇ when applied to 0-forms, the curl operator ∇× when applied to 1-forms, or the divergence operator ∇· when applied 2-forms. The nilpotent property dd=0 directly leads to the vector field analysis identities ∇×∇=0 and ∇·∇×=0.


A differential form ω∈Ω^k(M) is called closed if dω = 0, or exact if there is a (k-1)-form ζ∈Ω^k-1(M) such that ω = dζ. Due to the property dd=0, every exact form is closed. Thus, the differential d links the sequence of the spaces of differential forms on M into a chain complex

    0⟶Ω^0(M) d⟶Ω^1(M) d⟶⋯d⟶Ω^m-1(M) d⟶Ω^m(M)  0.

The k-th de Rham cohomology group, denoted by H^k_dR(M), is then defined to be the k-th homology of this chain complex, i.e., the quotient space of closed k-forms modulo the space of exact k-forms, i.e.,

    H^k_dR(M) = (d:Ω^k(M)→Ω^k+1(M))/(d:Ω^k-1(M)→Ω^k(M)).


The de Rham cohomology, by the de Rham theorem, is naturally isomorphic to the singular cohomology, and thus depends only on the manifold topology.



Let g be a Riemannian metric on M and ⟨·,·⟩_g be the pointwise inner product induced by g on Ω^k(M). The Hodge star operator ⋆ provides an isomorphism from the space of differential k-forms Ω^k(M) to the space of (m-k)-forms Ω^m-k(M), defined by the following formula

    ω∧⋆η = ⟨ω,η⟩_g μ_g,

where μ_g is the volume form on M induced by g. The Hodge L^2-inner product on the space of k-forms Ω^k(M) can then be obtained by taking the integral of the formula (<ref>)

    (ω, η) = ∫_Mω∧⋆η.

The codifferential δ: Ω^k(M)→Ω^k-1(M) is defined by

    δ = (-1)^m(k-1)+1⋆ d⋆,

which also has the nilpotent property δδ = 0. We call a differential form ω∈Ω^k(M) coclosed if δω = 0, or coexact if there is a (k+1)-form η∈Ω^k+1(M) such that ω = δη. 

The codifferential δ, as the differential d, also extends the classical gradient, curl and divergence in vector calculus. In ^3, it corresponds to -∇·, ∇× and -∇ when applied to 1-forms, 2-forms and 3-forms, respectively. 


The Hodge Laplacian for differential forms is defined as Δ = dδ + δ d: Ω^k(M)→Ω^k(M). Its kernel, consisting of all differential k-forms ω on M with Δω = 0, is called the space of harmonic k-forms.

We denote by ℋ^k_Δ(M) the space of harmonic k-forms and by ℋ^k(M) the space of k-forms that are both closed and coclosed, i.e., ℋ^k(M) =  d∩δ. The latter space ℋ^k(M), known as the space of harmonic k-fields, is in general only a subset of the space of harmonic forms ℋ^k(M)⊂ℋ^k_Δ(M), and is infinite-dimensional <cit.>. However, in the case of closed manifolds where ∂ M=∅, the space of harmonic forms ℋ^k_Δ(M) reduces to the space ℋ^k(M), as any harmonic form is both closed and coclosed. The result follows directly from the following formula

    0 = (Δω, ω) = ((dδ + δ d)ω, ω) =  (dω, dω) + (δω, δω),

due to the L^2-adjointness of the codifferential δ and the differential d on closed manifolds, i.e., (dω, η) = (ω, δη).

The classical Hodge decomposition theorem for closed manifolds states that the space of differential k-forms Ω^k(M) can be decomposed as

    Ω^k(M)= dΩ^k-1(M)⊕δΩ^k+1(M) ⊕ℋ^k_Δ(M).

These three subspaces are mutually orthogonal with respect to the inner product (<ref>). Moreover, Hodge theorem identifies the harmonic space ℋ^k_Δ(M) with the k-th de Rham cohomology group H^k_dR(M), which states that each harmonic form corresponds to exactly one equivalence class in H^k_dR(M). Therefore, the harmonic space ℋ^k_Δ(M) is fully determined by the manifold topology, and is finite-dimensional with its dimension given by the Betti number ℋ^k_Δ(M) = β_k.



















 §.§ Hodge decomposition for manifolds with boundary

In the presence of a non-empty boundary ∂ M, the two operators d and δ are not L^2-adjoint, as integration by parts leads to

    (dω, η) = (ω, δη) + ∫_∂ Mω∧⋆η,

which contains a boundary term that may not vanish, and thus the decomposed subspaces in (<ref>) are not orthogonal. However, certain boundary conditions can be enforced, ensuring the adjointness of the differential d and the codifferential δ, thereby inducing an orthogonal decomposition of the space of differential forms. 

The most common choices of boundary conditions ensuring the adjointness of d and δ are the normal (Dirichlet) and tangential (Neumann) boundary conditions. 

A differential form ω∈Ω^k(M) is called normal (Dirichlet) if it gives zero when applied to tangent vectors of the boundary, or tangential (Neumann) if the same holds for its dual ⋆ω instead. Denote by Ω^k_n(M) the set of normal differential k-forms and by Ω^k_t(M) the set of tangential differential forms, i.e.,

    Ω^k_n(M)    = {ω∈Ω^k(M) |  ω|_∂ M = 0}
    Ω^k_t(M)    = {ω∈Ω^k(M) |  ⋆ω|_∂ M = 0}.

Following their definitions, the spaces Ω^k_n(M) and Ω^m-k_t(M) are isomorphic under the Hodge star operator ⋆, also known as the Hodge duality. Moreover, the differential d preserves the normal boundary conditions, while the codifferential δ preserves the tangential boundary conditions.


The Hodge-Morrey decomposition <cit.> states that there is a 3-component L^2-orthogonal decomposition

    Ω^k(M) = dΩ^k-1_n(M)⊕δΩ^k+1_t(M) ⊕ℋ^k(M),

The orthogonality of the decomposition directly comes from the adjointness of δ and d when enforcing the normal or tangential boundary conditions. For ω∈Ω^k(M), there is a unique decomposition of ω given as follows:

    ω = dα_n + δβ_t + η,

where α_n∈Ω^k+1_n(M), β_t∈Ω_t^k+1(M), and η∈ℋ^k(M). Note that the potentials α_n and β_t are not uniquely determined as all α_n+dη and β_t+δγ with any η∈Ω_n^k-2(M) and
γ∈Ω_t^k+2(M) serve as potentials for the same components. However, the issue can be addressed by enforcing gauge conditions, such as


    δα_n = 0,
    dβ_t = 0.

The potentials α_n and β_t can then be uniquely determined by the following equations

    Δα_n = δω
    Δβ_t = dω,

by resolving the (finite) rank deficiencies of Δ under these boundary conditions. 


	In the case that M is a closed manifold, i.e., ∂ M = ∅, both the spaces Ω^k_n(M) and Ω^k_t(M) coincide with the space of differential forms Ω^k(M), and the space of harmonic fields is identical to the space of harmonic forms ℋ^k(M)=ℋ^k_Δ(M). The Hodge decomposition (<ref>) then reduces to the classical Hodge decomposition (<ref>) for closed manifolds.



	The Hodge-Morrey decomposition (<ref>) in the low dimensional Euclidean spaces ^2 and ^3, often referred to as the Helmholtz-Hodge decomposition in vector calculus, states that any vector field 𝐯 defined on a compact domain can be orthogonality decomposed as
	
    𝐯 = ∇ f + ∇×𝐮 + 𝐡,

	where f is a scalar potential that vanishes on the boundary of the domain, 𝐮 is a vector field orthogonal to the boundary, and 𝐡 is the harmonic vector field satisfying ∇×𝐡 = 0 and ∇·𝐡 = 0.
	The first component ∇ f and the second component ∇×𝐮 are often called the curl-free and divergence-free parts of the vector field 𝐯 respectively. Note that in the presence of a boundary, the resulting scalar potential f is also called satisfying the normal boundary of 0-forms, and the vector field 𝐮 is called satisfying the tangential boundary condition of 2-forms, which are direct counterparts of the potentials α_n and β_t in (<ref>). For a complete correspondence between scalar or vector fields, and differential forms under the normal and tangential boundary conditions, see <cit.>.


The space of harmonic fields ℋ^k, in general, is infinite-dimensional, and thus has no direct correspondence with the cohomology of the manifold. However, as noted by <cit.>, one can restrict to the space of normal harmonic fields, namely ℋ^k_n(M) = ℋ^k(M)∩Ω^k_n(M), and the space of tangential harmonic fields, ℋ^k_t(M)=ℋ^k(M)∩Ω^k_t(M). As a consequence of the de Rham map, these two subspaces ℋ^k_n(M) and ℋ^k_t(M) are fully determined by the topology of M: the space of normal harmonic fields ℋ^k_n(M) is isomorphic to the relative de Rham cohomology H^k_dR(M,∂ M), while the space of tangential harmonic fields ℋ^k_t(M) is isomorphic to the absolute de Rham cohomology H^k_dR(M) <cit.>. The two subspaces
ℋ^k_n(M) and ℋ^k_t(M) are thus finite-dimensional, with dimensions given by the Betti numbers: ℋ^k_n(M) = β_m-k and ℋ^k_t(M) = β_k. Furthermore, the kernels of the Hodge Laplacian Δ, when restricted to the space of normal forms Ω^k_n(M) and the space of tangential forms Ω^k_t(M) with gauge conditions on the boundary, can be identified to the space of normal harmonic fields and the space of tangential harmonic fields, respectively. Denote by Δ_n and Δ_t the restrictions of the Hodge Laplacian Δ on the space of normal fields Ω^k_n(M) satisfying Eq. (<ref>) and the space of tangential fields Ω^k_t(M) satisfying Eq. (<ref>), i.e., Δ_n:Ω^k_n(M)→Ω^k(M) and Δ_t:Ω^k_t(M)→Ω^k(M). Then immediately we have Δ_n = ℋ^k(M)∩Ω^k_n(M)=ℋ^k_n(M) and Δ_t = ℋ^k(M)∩Ω^k_t(M) = ℋ^k_t(M). The result follows directly from Eq. (<ref>). These identifications, finally, enable us to study the topology of the underlying manifold M through the Hodge Laplacians on normal and tangential forms.


	In fact, let ℋ^k_co = ℋ^k(M)∩δΩ^k+1(M) and ℋ^k_ex = ℋ^k(M)∩ dΩ^k-1(M). The space of harmonic fields ℋ^k(M) can be further orthogonally decomposed for smooth manifolds
	
    ℋ^k(M) =    ℋ^k_co(M)⊕ℋ^k_n(M) 
     =    ℋ^k_ex(M)⊕ℋ^k_t(M),

which results in the Hodge-Morrey-Friedrichs decomposition given as follows

    Ω^k(M)    = dΩ^k-1_n(M)⊕δΩ^k+1_t(M) ⊕ℋ^k_co(M)⊕ℋ^k_n(M)
        = dΩ^k-1_n(M)⊕δΩ^k+1_t(M) ⊕ℋ^k_ex(M)⊕ℋ^k_t(M).

In particular, if M is a compact domain in Euclidean spaces, then there is a unique orthogonal 5-component decomposition 

    Ω^k(M) = dΩ^k-1_n(M)⊕δΩ^k+1_t(M) ⊕ℋ^k_n(M) ⊕ℋ^k_t(M)⊕ (dΩ^k-1(M)∩δΩ^k+1(M)),

as the spaces ℋ^k_n(M) and ℋ^k_t(M) are  L^2-orthogonal, instead of just being transversal for compact manifolds in general <cit.>. Due to the correspondence between differential forms and vector fields in the low-dimensional Euclidean spaces, the implementation of this 5-component Hodge decomposition has been applied and implemented to the study of vector fields for surface triangle meshes, for tetrahedral meshes <cit.> and for regular Cartesian grids <cit.>.







As we mainly focus on applications of compact domains in ^3, to study the geometric and topological information of the underlying manifolds, there are eight Laplacians to be considered, which are defined on the spaces of differential k-forms with k= 0,1,2,3 satisfying either the normal or the tangential boundary conditions. However, thanks to the duality between the space of normal fields and tangential fields, the study of the spectra of these eight Laplacians reduces to that of four Laplacians on one of the two types of boundary conditions, and finally to the singular spectra of three differential operators, applied to differential forms of degree k = 0,1,2,3 <cit.>. Further details will be discussed in the next section for the discretization of Laplacians.





§ SPECTRAL ANALYSIS


In this section, we elaborate on the discretization of the Hodge Laplacian and introduce the Boundary-Induced Graph (BIG) Laplacian for compact domains in low-dimensional Euclidean spaces. Although the theory works for 2D compact domains, we focus only, for the remainder of the paper, compact domains in ^3, as we target mainly 3D applications. We use DEC to discretize all differential operators and differential forms on regular Cartesian grids, as it allows for efficient and accurate numerical algorithms relying on just matrix algebra, while keeping the L^2 orthogonality between different components in Hodge decomposition. In addition, the constructed discrete differential operators and differential forms in DEC approximate their smooth analogs. For the characterization of the underlying manifold, we choose the Eulerian formulation, where the manifold is given as a sublevel set of a level set function defined on a regular Cartesian grid. Another common way, called the Lagrangian formulation, discretizes the manifold as simplicial meshes, i.e., triangular or tetrahedral meshes in 2D or 3D. The spectrum analysis of the Hodge Laplacians has been discussed in <cit.> for the Lagrangian formulation and in <cit.> for the Eulerian formulation. Compared to the Lagrangian case, the Eulerian representation uses vertices, edges, faces and cells all fixed in a Cartesian grid, which significantly simplifies the data structures and algorithms. The Hodge stars, in the latter case, are close to rescaled identity matrices. This fact simplifies the study of Hodge Laplacians to that of BIG Laplacians with no Hodge stars involved, and thus leads to algorithms with efficient computations. 




 §.§ Discretization on entire grid

Denote by I_m a rectangular m-dimensional regular Cartesian grid with k-cells oriented according to their alignments with the coordinate axes. The entire grid I_m can be treated as a cell complex tessellating a rectangular domain in ^m, where each k-cell is a k-dimensional hypercube with edge length ℓ. A continuous differential k-form ω on I_m, following the de Rham map, can be discretized by its integral value over each oriented k-cell σ_i, given as W^i=∫_σ_iω <cit.>. The discrete differential on discrete k-forms of the grid I_m is then encoded by a sparse matrix D^I_k, which stores the signed incidence between (k+1)-cells and k-cells and is given as the transpose of the cell boundary operator ∂_k+1^T on (k+1)-cells following from Stokes' theorem ∫_σ dω= ∫_∂σω. An illustration of the chain complex formed by boundary operator ∂ for a simple grid complex with a single 2D cell can be seen in Fig. <ref>, which is a straightforward generalization of the chain complex on simplicial complexes. Note that the boundary of the boundary of a cell always results in a 0 chain, i.e., ∂∂ = 0, whose transpose immediately produces D^I_k+1D^I_k = 0, thus preserving the nilpotent property in the continuous setting.



The discrete Hodge star establishes a one-to-one correspondence between discrete k-forms on the primal grid I_m and discrete (m-k)-forms on its dual grid, given as the translated grid with grid points located at the m-cell centers of I_m, based on the following formula

    1/|σ_k|∫_σ_kω≈1/|⋆σ_k|∫_⋆σ_k⋆ω,

where ⋆σ_k is the dual (m-k)-cell formed by the dual grid points located at the centers of the primal m-cells incident to σ_k. See Fig. <ref> for an illustration for the correspondences between the primal and dual cells in the Cartesian grid case. Following from the discretization of differential forms, this correspondence leads to a diagonal matrix S^I_k with diagonal entries given by the ratio between the volumes of the dual  (m-k)-cells and the primal k-cells, ℓ^m-k/ℓ^k=ℓ^m-2k. The associated discrete Hodge L^2-inner product (<ref>) of two discrete k-forms V_k and W_k on grid I_m is then given by

    (V_k, W_k)^I = V_k^TS^I_kW_k.

The discrete codifferential, by definition of its smooth counterpart (<ref>), can be assembled from the discrete differential and Hodge star operators as δ^I_k = (S^I_k-1)^-1D^I_k-1S^I_k. Note that the discrete counterpart of the Hodge Laplacian Δ = dδ + δ d by replacing the differential and codifferential operators results in a nonsymmetric matrix. Instead, we consider the counterpart of ⋆Δ as the discrete Hodge Laplacian given by

    L^I_k = (D^I_k)^TS^I_k+1D^I_k + S^I_kD^I_k-1(S^I_k-1)^-1(D^I_k-1)^TS^I_k,

where the operators are considered to be null for k<0 or k>m.






 §.§ Discrete differential forms and operators on M



 
Compared to the case of simplicial or polygonal meshes, where the projection matrices to the interior can be straightforward to implement with the boundary elements explicitly labeled, modeling the manifold M as the volume bounded by a level set surface leads to delicate computation of the projection matrices. Note that the boundary of M using grid representation typically intersects with boundary k-cells instead of being its supersets. We restrict the computation to relevant cells by implementing the two types of boundary conditions through the inclusion or exclusion of the entire k-cells. We use the strategy as in <cit.> for the computation of projection matrices for each type of boundary condition: for the normal boundary condition, we include all cells if at least one of its vertices is inside or on the boundary of M, while for the tangential boundary condition, we include all cells with at least one of the vertices of the corresponding dual cells is inside or on the boundary. We refer to the former set of cells as the normal support and the latter as the tangential support. In contrast to the mesh case, it is important to note that neither the normal nor the tangential support is necessarily a superset of the other. See Fig. <ref> for one example showing the distinction of these two supports for 1-forms. 


In the computation of the discrete Hodge star operators, it is essential to consider and incorporate the boundary conditions. Following the procedure in <cit.>, we keep the dual cell volumes and adjust the primal cell volumes for normal boundary conditions, and do conversely for tangential boundary conditions with the primal cell volumes kept and the dual cell volumes changed. To be specific, when dealing with normal (resp. tangential) boundary conditions, we only compute the volume of the region of the primal (resp. dual) k-cells within the boundary ∂ M for the denominator (resp. numerator) of the ratio in the discrete Hodge star matrix, and leave the dual (resp. primal) cell volumes in the numerator (resp. denominator) unchanged. Each unaltered k-cell has a k-volume of ℓ^k. In addition, For numerical stability, we do not alter the volume of outside primal k-cells, and perturb the level set function evaluated at primal/dual grid points to have an absolute value above ϵ=10^-5ℓ, which ensures well-behaved fractional k-volumes. We denote by S^I_k,n and S^I_k,t the diagonal Hodge star matrices defined on the entire grid I^m corresponding to the normal and tangential boundary conditions, respectively.

The projection matrix to the corresponding support, for each type of boundary condition, can be constructed from the identity matrices by eliminating the rows corresponding to k-cells outside the support. Denote by P_k,n the projection matrix for k-cells onto the normal support and by P_k,t the one onto the tangential support. We then obtain a new set of differential and Hodge star operators for M:

    D_k,n   = P_k+1,nD_kP_k,n^T,   S_k,n = P_k, nS^I_k,nP_k,n^T
    
    	D_k,t   = P_k+1,tD_kP_k,t^T,   S_k,t = P_k, tS^I_k,tP_k,t^T

The nilpotent property D_k+1,nD_k,n = 0 and D_k+1,tD_k,t = 0 still holds for both boundary conditions due to D^I_k+1D^I_k=0 and the following observations

    P_k+1,n^T P_k+1, nD^I_k P_k, n^T = D^I_k P_k, n^T,  
    	P_k+1, t D^I_k P_k, t^T P_k,t = P_k+1,t D^I_k.

The discrete Hodge L^2-inner products of the two types of discrete k-forms on the manifold M for these two boundary conditions are then given by

    (ξ^k, ζ^k)^n    = (ξ^k)^TS_k,nζ^k 
    
    	(ξ^k, ζ^k)^t    = (ξ^k)^TS_k,tζ^k,

whose domains are the discrete Ω^k_n(M) and the discrete Ω^k_t(M) respectively. Finally, we assemble the two types of discrete Hodge Laplacians as in the mesh case:

    L_k, n   = D_k, n^TS_k+1, nD_k, n + S_k, nD_k-1, nS_k-1, n^-1D_k-1, n^TS_k, n
    
    	L_k, t   = D_k, t^TS_k+1, tD_k, t + S_k, tD_k-1, tS_k-1, t^-1D_k-1, t^TS_k, t.

The null spaces of these discrete Hodge Laplacians, as in the continuous case, are fully determined by the topology of the underlying manifold M, since they only depend on the differential and projection matrices. The dimension of the kernel of L_k, n is given by the Betti number β_m-k, while the dimension of the kernel of L_k, t is given by β_k. Here the Betti number β_k presents directly the number of k-dimensional holes on the manifold M. For instance, β_0 gives the number of connected components, β_1 gives the number of tunnels, and β_2 provides the number of closed cavities, respectively. The spectra of these Laplacians, in addition, could be used to study the geometric information of the manifold. It is known that the non-zero eigenvalues of the Laplacians provide rich insights into the shape of a manifold. For instance, the Fiedler value, defined as the smallest non-zero eigenvalue of a graph Laplacian, describes connectivity. As another example, the multiplicity of eigenvalues can reveal certain symmetries of the shape.


	The two types of discrete Hodge Laplacians (<ref>) not only provide rich geometrical and topological information of the underlying manifold, but also play a central role in the computation of the discrete Hodge decomposition (<ref>) of differential forms for compact domains in 2D and 3D Euclidean spaces. In particular, they can be utilized, by resolving the rank deficiencies, to compute the potentials of the decomposed components in Hodge decomposition on normal or tangential support satisfying the corresponding boundary conditions. In addition, as the kernel sizes of Laplacians are finite, their eigenvectors corresponding to 0 eigenvalues, for each k, form a basis for the space of normal or tangential harmonic fields.



Note that the discrete Hodge stars in the Eulerian setting are almost identical to rescaled identity matrices. Therefore, the computations of the Hodge Laplacian can be further simplified by replacing the Hodge stars with identity matrices, leading to the definition of the Boundary-Induced Graph (BIG) Laplacians as follows:

    L^B_k, n   = D_k, n^TD_k, n + D_k-1, nD_k-1, n^T
    
    	L^B_k, t   = D_k, t^TD_k, t + D_k-1, tD_k-1, t^T.

The BIG Laplacians were introduced in <cit.> for bounded domains to facilitate the comparison and contrast of the Hodge Laplacians and the combinatorial Laplacians. They preserve the Hodge Laplacian's capability to perform differential calculus but also retain the discrete nature of combinatorial Laplacians. The convergence of the spectra of the BIG Laplacians to Hodge Laplacians has been discussed in <cit.>, showing that the spectra of (<ref>) converge to those of Hodge Laplacians up to a scaling value ℓ^-2 when enforcing the boundary conditions. This scaling value ℓ^-2 is exactly the ratio between the missing scaling factor ℓ^m-2(k+1) in L_k and the missing factor ℓ^m-2k of S_k. As the BIG Laplacians produce results similar to those obtained from the discrete Hodge Laplacians with less computation, they can also be used to study the geometric and topological information of the underlying manifolds.


Note that the dual grid is also a Cartesian grid staggered with the primal grid by a replacement of ℓ/2 in all three axial directions of the Cartesian coordinates. For the study of the spectra of these Laplacians, one only needs to implement one type of boundary condition, for instance, the normal boundary condition, as L_k,n defined on the primal grid with normal boundary conditions is equivalent to L_m-k,t defined on its dual grid with tangential boundary conditions.



 §.§ Spectral analysis of Laplacians



Note that, on the grid, the Hodge Laplacians and the BIG Laplacians are of the same sparsity patterns. For simplicity in exposition when discussing the spectrum analysis of the Laplacians, we let L_k be a generic Laplacian matrix of the form

    L_k   = D_k^TS_k+1D_k + S_kD_k-1S_k-1^-1D_k-1^TS_k.

Here the Laplacian L_k can be interpreted, under choices of boundary conditions and Hodge star accuracy, as either a Hodge Laplacian, or BIG Laplacian ( with S_k set to identity), under tangential or normal boundary condition. The eigenvalues and eigenvectors of L_k can be solved by considering the generalized eigenvalue problem

    L_k W =λ S_k W,

where λ is an eigenvalue, and W is the associated eigenvector.
To analyze the results, we perform the following transformation in the space of discrete forms: D̅_k = S_k+1^1/2D_kS_k^-1/2, L̅_k = S_k^-1/2L_k S_k^-1/2 and W̅ = S_k^1/2W. Rewriting the formulas above yields a simplified form of the Laplacian

    L̅_k = D̅_k^TD̅_k + D̅_k-1D̅_k-1^T,

and a regular eigenvalue problem:

    L̅_k W̅ = λW̅.

Note that the property D̅_kD̅_k-1 = 0 is preserved.





As the non-zero eigenvalues of D̅_k^TD̅_k and D̅_kD̅_k^T for each k are the same, given by the squared non-zero singular values of the discrete differential D̅_k, and each Laplacian L̅_k is just the combination of D̅_k^TD̅_k and D̅_k-1D_k-1^T, the entire spectrum of the Laplacians can thus be studied through the singular values of discrete differentials. Let 

    D̅_k = U_k+1Σ_kV_k^T

be the singular value decomposition of D̅_k, where U_k+1 and V_k are orthogonal matrices and Σ_k is a rectangular diagonal matrix with diagonal values given by the singular values of D̅_k. It follows immediately from D̅_kD̅_k-1 = 0 that 

    Σ_kV_k^TU_kΣ_k-1 = 0.

Therefore, the columns of V_k corresponding to non-zero singular values of D̅_k are orthogonal to columns of U_k associated with non-zero singular values of D̅_k-1. In addition, it follows from

    L_k = V_kΣ_k^2V_k^T + U_kΣ_k-1^2U_k^T

that the spectrum of L̅_k is given by the union of squared non-zero singular values of D̅_k and D̅_k-1, and 0, with the multiplicity of 0 given by the k-th Betti numbers. The columns of U_k and V_k corresponding to non-zero singular values, together with the set of harmonic forms, span the entire space of differential k-forms. 

In the case that (M) = 3, for each type of boundary condition, we have four Laplacians of different degrees in total k = 0,1,2,3:

    L̅_0   = D̅_0^TD̅_0
    L̅_1   = D̅_1^TD̅_1 + D_0D_0^T
    L̅_2   = D̅_2^TD̅_2 + D_1D_1^T
    L̅_3   = D̅_2D̅_2^T.

Due to the aforementioned discussion on the spectrum of Laplacians and the duality of the normal and tangential boundary conditions, the spectral analysis of all Laplacians can be reduced to the singular spectra analysis of the three discrete differentials D̅_0, D̅_1, and D̅_2 with one type of boundary conditions. Note that the numerical evaluation of the singular values of these differentials, in the simplicial mesh case, may differ for the two types of boundary conditions, as the DoF for normal k-forms and tangent m-k forms are different. However, in the Cartesian representation, they are strictly equivalent to each other by shifting the grid in all directions of the axis by ℓ/2, so long as M is at least one grid spacing away from the boundary of the grid. 

For the computation of the spectra of the Laplacians, we choose the normal boundary condition. The spectra of all Laplacians L̅_k,n for compact domains in ^3 can be finally decomposed into three distinct parts: the squared singular values of the gradient of tangential scalar fields, denoted by T, the squared singular values of the gradient of normal scalar fields, denoted by N, and the squared singular values of the curl of tangential curl fields, denoted by C.




§ PERSISTENT DE RHAM-HODGE LAPLACIANS


In this section, we present the construction of the persistent de Rham-Hodge Laplacian on differentiable manifolds, which is based on the filtration of manifolds induced by varying a single parameter (the filtration parameter). The spectra of Laplacians carry rich topological and geometric information of a manifold. However, a single manifold might not provide enough information in applications like feature extraction for machine learning analysis. As such, instead of studying just a single manifold, one could examine the spectra of a family of manifolds by adjusting the filtration parameter. The spectra of the Laplacians from this family of manifolds could provide much more information than by considering just one, as the topology and geometry could change for different parameters. This single-parameter family of manifolds, called the evolution of manifolds, was first introduced in <cit.> based on tetrahedral meshes. We briefly recap the background. 

The formal definition of the evolving manifold is given by a one-parameter family of immersions F_c = F(·, c) with F: B× [a,b]→ N being a smooth map, where B is called the base manifold, N is the ambient manifold, and c∈[a,b] is a real parameter within the interval. In practice, the most common way to define the evolution of manifolds without specifying B is through a level set function by adjusting the isovalues. Given a function f: N→ [a,b], then in our case, we consider the sublevel sets M = {x∈ N |  f(x)≤ c} with boundary given as ∂ M = {x∈ N |  f(x) = c} for c∈ [a,b]. A sequence of manifolds can then be obtained by considering evenly distributed isovalues of the function f with the inclusion map

    M_0 M_1 M_2⋯ M_s-1 M_s,

where each M_l is given as the sublevel set corresponding to c_l with a≤ c_0<c_1<⋯ < c_s ≤ b. To ensure that M_l is a manifold, we assume that the function f is a Morse function on N, and none of the c_l's corresponds to a critical value of the function f, i.e., f^-1(c_l) does not contain any critical points. This is always possible as the set of Morse functions on a compact manifold is dense in the space of smooth functions, and their critical points are isolated, nondegenerate, and finite for compact manifolds. Thus we can always perturb any input function slightly to avoid critical isovalues in {c_l, l =0, 1, ⋯ s}. In addition, we assume that for each l, M_l, l+1 = M_l+1\ M_l={x∈ N|  f(x)∈[c_l,c_l+1]} contains at most one critical point, which can be realized by refining the parameter sequence. Note that both M_l and M_l, l+1 are compact. By Morse theory, if M_l, l+1 contains no critical points, M_l is diffeomorphic to M_l+1. The retraction from M_l+1 to M_l can be easily constructed by considering a flow along the gradient of the function. As M_l+1 is homotopic to M_l in this case, there is no topological change happening between (c_l, c_l+1). For the other case when there is exactly one critical point in M_l, l+1, the manifold M_l+1 is homotopic to M_l with a k-cell attached, where k is the index of the critical point, defined to be the dimension of the largest subspace on which the Hessian Hess(f)(x) is negative definite. The topological change of the sublevel sets occurs precisely at the critical values of the level set function. Depending on the type of the critical points, i.e., local minimum, saddle points, and local maximum, the topology changes in different ways. In general, a local maximum has the full index m, a local minimum has index 0, while saddle points have indices strictly between 0 and m. In the case of ^3, the occurrences of minima and maxima correspond to the birth of the 0-th generators and the death of the 2nd homology generators respectively, while the occurrences of 1-saddle points correspond to the birth of 1st homology generators or the death of the 0-th homology generators, and those of 2-saddle points correspond to the birth of 2nd homology generators or the death of 1st homology generators.



 §.§ Persistent harmonic forms

As the de Rham complex depends on the topology, it can also be extended to the filtration of manifolds. Due to the duality of the normal and tangential boundary conditions, without loss of generality, one may focus on the space of normal differential forms. Given M_l M_l+1, we then need to construct a map from the space of normal k-forms Ω^k_n(M_l) to the space of normal k-forms Ω^k_n(M_l+1), that extends each normal k-form on M_l to a normal k-form on M_l+1. Let ω∈Ω^k_n(M_l). The idea is to utilize the boundary condition of ω on M_l and extend the forms ω|_∂ M_l to exact normal forms on the domain M_l, l+1 with certain boundary conditions on ∂ M_l, l+1 = ∂ M_l∪∂ M_l+1. Then the combination ω defines a normal k-form on the manifold M_l+1. Note however that δω is only 0 in M_l, so the extension of ω∈δ may no longer be in δ on M_l+1.

To be specific, we consider the biharmonic equation Δ^2ζ=Δ(Δζ) = 0 on M_l, l+1 with both Dirichlet and Neumann boundary conditions to ensure the smoothness of d ζ with ω through ∂ M_l. Note that d ζ satisfies the normal boundary condition on M_l,l+1. Let ω be the extension of ω on M_l+1 with ω = ω on M_l and ω = d ζ on M_l, l+1. It follows that ω∈Ω^k_n(M_l+1) as it satisfies the normal boundary condition ω|_∂ M_l+1 = ζ|_∂ M_l+1=0. 

While the biharmonic equation produces a smooth extension, in practice, it is more efficient to consider the harmonic extension with the boundary condition Δζ = 0 with the boundary condition of ⋆ dζ = ⋆ω on ∂ M_l and the typical normal form boundary condition on ∂ M_l+1. The solution, by <cit.>, is unique. The resulting ω̅ is continuous but nonsmooth as δω̅ may lead to a Dirac distribution on ∂ M_l when M_l,l+1 induces a topological change. For instance, for a harmonic normal 1-form ω on a spherical shell M_l with M_l+1 turning into a solid ball, the biharmonic extension would create a uniform divergence δω̅ in M_l,l+1, whereas the harmonic extension creates a thin layer of nonzero divergence δω̅ near the part of ∂ M_l around the cavity in the middle. Thus, the harmonic extension serves the same purpose in reducing the kernel of δ.

Denote by ℐ_l, 1 the map from Ω^k_n(M_l) to Ω^k_n(M_l+1) sending ω to ω.  Note that (d∘ℐ_l, 1)(ω) is 0 on M_l,l+1 and thus the same as the extension of the differential of a normal form dω on M_l, l+1, i.e., d∘ℐ_l, 1 =ℐ_l, 1∘ d. It follows that there is a commutative diagram

    Ω^0_n(M_0) rddℐ^0_0,1   Ω^1_n(M_0) rddℐ^1_0,1   Ω^2_n(M_0) rddℐ^2_0,1   Ω^3_n(M_0) dℐ^3_0,1
    Ω^0_n(M_1) rddℐ^0_1,1   Ω^1_n(M_1)  rddℐ^1_1,1   Ω^2_n(M_1)  rddℐ^2_1,1   Ω^3_n(M_1)  dℐ^3_1,1
    Ω^0_n(M_2)  rddℐ^0_2,1   Ω^1_n(M_2) rddℐ^1_2,1   Ω^2_n(M_2) rddℐ^2_2,1   Ω^3_n(M_2) dℐ^3_2,1
    ⋯   ⋯   ⋯   ⋯

where the horizontal direction gives the de Rham complex and the vertical direction shows the filtration-induced extensions.


Next, we introduce the p-persistent Hodge Laplacian. Let ℐ_l, p = ℐ_l+p-1, 1∘ ...∘ℐ_l, 1, which then gives an extension map from the space of normal forms on M_l to the space of normal forms on M_l+p. We have the following commutative diagram

    Ω_n^k(M_l)      Ω_n^k+1(M_l)
    
    Ω_n^k-1(M_l+p)      Ω_n^k(M_l+p)      ["d_l+p^k-1", shift left, from=3-1, to=3-3]
    	["R_l,p", shift left, from=3-3, to=1-3]
    	["δ_l+p^k", shift left, from=3-3, to=3-1]
    	["d_l^k", shift left, from=1-3, to=1-5]
    	["δ̃_l,p^k", shift left=-2, from=1-3, to=3-1]
    	["I_l,p", shift left, from=1-3, to=3-3]
    	["δ_l^k+1", shift left, from=1-5, to=1-3]
    	["d̃_l,p^k-1", shift left=4, from=3-1, to=1-3]
    .

















































Here d_l, δ_l denotes the differential and codifferential on Ω^k(M_l), d_l+p, δ_l+p denotes the differential and codifferential on Ω^k(M_l+p), respectively, and ℛ_l, p is the projection of differential forms in Ω^k_n(M_l+p) to the space spanned by the harmonic extensions followed by the restriction to M_l. Let δ̃_l,p = δ_l+p∘ℐ_l, p and d̃_l,p = ℛ_l, p∘ d_l+p. By the construction of the extension, we have 
(δ̃_l,pω, η) = (ω, d̃_l,pη), i.e., δ̃_l,p are adjoint to d̃_l,p. 
We then define the p-persistent Hodge Laplacian operator Δ^p_n, l: Ω^k_n(M_l)→Ω^k_n(M_l) as follows

    Δ^p_n, l = d̃_l,pδ̃_l,p + 
    	δ_ld_l.

It is easy to see that when p=0, the p-persistent Hodge Laplacian gives exactly the usual Hodge Laplacian Δ_n, l: Ω^k_n(M_l)→Ω^k_n(M_l) restricted to the space of normal forms. We then define the p-persistent normal harmonic fields as the kernel of the p-persistent Hodge Laplacian ℋ_n^k,p = Δ^p_n, l, which can be identified with the space δ̃_l,p∩ d_l. Note that by the extension construction and ℛ_l, p∘ℐ_l, p = Id, one can see that δ̃_l,p⊂δ gets smaller as p increases, which confirms that fewer cohomology generators persist longer. 



 §.§ Discretization of p-persistent de Rham cohomology



The regular Cartesian grid allows one to define persistent graph Laplacian on manifolds in the same way as persistent graph Laplacian <cit.>. It also allows defining persistent Hodge Laplacian in a consistent way, with the inclusion of nontrivial Hodge stars.

Recall that the discrete differential k-forms can be seen as a k-cochain, i.e., a linear mapping from the chain space 𝒞_k to  that sends a k-chain c_k = ∑_i a_iσ_i to ∫_c_kω = ∑_i a_iW_i, where W_i = ∫_σ_iω is the integral of a smooth k-form ω over the k-cell σ_i.


By varying the isovalue of the level set function f, we can get a sequence of cell complexes given as nested sequences of sub-cell complexes of K satisfying the normal boundary conditions.




    ∅ = K_0⊂ K_1 ⊂⋯⊂ K_s-1⊂ K_s = K.

See Fig. <ref> for an example of such a nested sequence of sub-cell complexes in a 2D Cartesian grid. Denote by 𝒞^k(K_l) the space of discrete k-forms on subcomplex K_l with 0≤ l≤ s. Note that K_l⊂ K_l+1. A discrete k-form on K_l can be easily extended to K_l+1 by solving the discrete Laplace equation with the above boundary conditions for values on every k-cells in K_l,l+1=Cl (K_l+1\ K_l), the closure of the difference complex. We denote this extension map as I_l, 1: 𝒞^k(K_l)→𝒞^k(K_l+1) and by I_l, p = I_l+p-1, 1∘ I_l+p-2, 1∘⋯∘ I_l, 1: 𝒞^k(K_l)→𝒞^k(K_l+p) the extension mapping from the space of discrete k-forms on K_l to the space of discrete k-forms on K_l+p, which may also be constructed directly by solving the Laplace equation on K_l,l+p=Cl(K_l+p\ K_l). With this extension mapping, the space of discrete k-forms on K_l can be seen as a subspace of discrete k-forms on K_l+p.

A sequence of the discrete de Rham cochain complexes can be defined as follows:

    𝒞^0(K_0)   𝒞^1(K_0)   ⋯   𝒞^k(K_0)   𝒞^k+1(K_0)   ⋯
    𝒞^0(K_1)   𝒞^1(K_1)   ⋯   𝒞^k(K_1)   𝒞^k+1(K_1)   ⋯
    𝒞^1(K_2)   𝒞^1(K_2)   ⋯   𝒞^k(K_2)   𝒞^k+1(K_2)   ⋯
    ⋯   ⋯      ⋯   ⋯["D_0^0", shift left, from=1-1, to=1-2]
    	["I_0,1", shift left, from=1-1, to=2-1]
    	["δ^1_0", shift left, from=1-2, to=1-1]
    	["D_0^1", shift left, from=1-2, to=1-3]
    	["I_0,1", shift left, from=1-2, to=2-2]
    	["δ_0^2", shift left, from=1-3, to=1-2]
    	["D_0^k-1", shift left, from=1-3, to=1-4]
    	["δ_0^k", shift left, from=1-4, to=1-3]
    	["D_0^k", shift left, from=1-4, to=1-5]
    	["I_0,1", shift left, from=1-4, to=2-4]
    	["δ_0^k+1", shift left, from=1-5, to=1-4]
    	["D_0^k+1", shift left, from=1-5, to=1-6]
    	["I_0,1", shift left, from=1-5, to=2-5]
    	["δ_0^k+2", shift left, from=1-6, to=1-5]
    	["D_1^0", shift left, from=2-1, to=2-2]
    	["I_1,1", shift left, from=2-1, to=3-1]
    	["δ_1^1", shift left, from=2-2, to=2-1]
    	["D_1^1", shift left, from=2-2, to=2-3]
    	["I_1,1", shift left, from=2-2, to=3-2]
    	["δ^2_1", shift left, from=2-3, to=2-2]
    	["D_1^k-1", shift left, from=2-3, to=2-4]
    	["δ_1^k", shift left, from=2-4, to=2-3]
    	["D_1^k", shift left, from=2-4, to=2-5]
    	["I_1,1", shift left, from=2-4, to=3-4]
    	["δ_1^k+1", shift left, from=2-5, to=2-4]
    	["D_1^k+1", shift left, from=2-5, to=2-6]
    	["I_1,1", shift left, from=2-5, to=3-5]
    	["δ_1^k+2", shift left, from=2-6, to=2-5]
    	["D_2^0", shift left, from=3-1, to=3-2]
    	["I_2,1", shift left,from=3-1, to=4-1]
    	["δ_2^1", shift left, from=3-2, to=3-1]
    	["D_2^1", shift left, from=3-2, to=3-3]
    	["I_2,1", shift left, from=3-2, to=4-2]
    	["δ_2^2", shift left, from=3-3, to=3-2]
    	["D_2^k-1", shift left, from=3-3, to=3-4]
    	["δ_2^k", shift left, from=3-4, to=3-3]
    	["D_2^k", shift left, from=3-4, to=3-5]
    	["I_2,1", shift left, from=3-4, to=4-4]
    	["δ_2^k+1", shift left, from=3-5, to=3-4]
    	["D_2^k+1", shift left, from=3-5, to=3-6]
    	["I_2,1", shift left, from=3-5, to=4-5]
    	["δ_2^k+2", shift left, from=3-6, to=3-5]

where D_l^k: 𝒞^k+1(K_l)→𝒞^k(K_l) denotes the discrete differential operator, and δ_l^k: 𝒞^k(K_l)→𝒞^k-1(K_l) denotes the discrete codifferential operator on K_l.

To define the persistent discrete Hodge Laplacian, we construct the discrete counterparts of d̃_l,p and δ̃_l,p in the previous section. 

Denote by δ_l,p^k+1,n:𝒞^k+1(K_l)→𝒞_l,p^k  the operator given as δ_l,p^k,n = δ_l+p^k I_l,p^k,n, where δ_l+p^k,n is the previously defined discrete operator for K_l+p and I_l,p^k,n is the discrete harmonic extension operator defined next. Assuming K_l,l+p contains few  k-cells, the harmonic extension is then constructed by the linear system L^k-1,n_K_l,l+pζ = 0, and shifting all ⋆ dζ values in the overlap of supports of K_l and K_l,l+p to the right-hand side and replacing them with a rescaling of ⋆ω based on the k-volume within each support. More specifically, the resulting system is L̃^k-1,n_K_l,l+pζ̃= -S^k-1,nδ^k_∂ K_lω, where L̃^k-1,n_K_l,l+p is the Laplace operator applied to a form ζ̃ defined on K_l,l+p\∂ K_l, and δ^k_∂ K_l is the boundary codifferential operator that uses the values of ω on ∂ K_l to evaluate the neighboring (k-1)-cells in K_l,l+p\∂ K_l. 

The resulting extension operator 

    I_l,p= 
    [                                         Id_K_l; -D^k_K_l,l+p(L̃^k,n_K_l,l+p)^-1 S^k,nδ^k_∂ K_l ],

where Id_K_l is the identity matrix in K_l up to a rescaling in the boundary, 
provides the combination of ω in K_l and  dζ̃ in K_l,l+p\∂ K_l, when applied to ω. The matrix corresponding to I_l,p is dense for rows corresponding to cells in K_l,l+p but diagonal for rows corresponding to cells in K_l.  Note that δ_∂ K_l is not necessarily 0 for coclosed ω, but is 0 for coexact ω.

The adjoint operator of δ_l,p^k+1,n defines D^k_l,p. In the following, we drop most of the subscripts for clarity. Recall that (ω, d̃η)=(δ̃ω,η) can be discretized as

    [W]^T S [D̃ E]= [S^-1 D^T S I_l,p W]^T S [E]

with W and E as discrete versions of ω and η. Thus D̃= S^-1 I_l,p^T S D, from which we may recognize the restriction operator as R=S^-1I_l,p^T S. This restriction operator can be seen as the L_2-projection onto the space formed by all harmonic extensions from Ω_n^K(M_l). 








Note that in this case, we immediately δ_l,p^kδ_l^k+1 = 0, since the extension operator will generate ζ̃=0 for any coexact form ω=δβ on K_l as the right-hand side of the associated linear system essentially corresponds to δδβ=0. From the adjoint version, we have  D_l^kD_l,p^k-1 = 0, and thus the following commutative diagram
















    𝒞^k(K_l)      𝒞^k+1(K_l)
    
    𝒞^k-1(K_l+p)      𝒞^k(K_l+p)      ["D_l+p^k-1", shift left, from=3-1, to=3-3]
    	["R_l,p", shift left, from=3-3, to=1-3]
    	["δ_l+p^k", shift left, from=3-3, to=3-1]
    	["D_l^k", shift left, from=1-3, to=1-5]
    	["δ_l,p^k", shift left=-2, from=1-3, to=3-1]
    	["I_l,p", shift left, from=1-3, to=3-3]
    	["δ_l^k+1", shift left, from=1-5, to=1-3]
    	["D_l,p^k-1", shift left=4, from=3-1, to=1-3]
    .


The discrete p-persistent Hodge Laplacian is then given as follows 

    L_l,p^k = D_l,p^k-1δ_l,p^k + δ_l^k+1D_l^k,

and the discrete p-persistent BIG Laplacian is  

    L_l,p^k = D_l,p^k-1(D_l,p^k-1)^T + (D_l^k)^TD_l^k.














We now present some examples of evolving manifolds and show results for the spectral analysis of their persistent Laplacians. In particular, we focus on the changes of Betti numbers β_0, β_1 and β_2 and the first non-zero eigenvalues λ^T_1, λ^C_1 and λ^N_1 of the 0-persistent BIG Laplacians in the set T, C and N, respectively, as introduced in Sec. <ref>. Four models are considered, including the Bimba model, the kitten model, a genus-3 model, and a four-ball model. For each model, we show on the top row snapshots of evolving manifolds at five evenly spaced isovalues in a chosen interval, and on the bottom row the changes in Betti numbers and the first non-zero eigenvalues λ^T_1, λ^C_1 and λ^N_1. All the evolving manifolds are generated using isovalues of the signed distance function (SDF) from the original surface model, given as the 0-isosurface of the SDF. As we show below, these values from the evolution of manifolds provide rich information than considering just a single manifold. The discontinuity of these variables indicates the topological changes occurring during the evolution process, and the monotonicity of these non-zero eigenvalues reveals the geometric changes. 


The results for the Bimba model are presented in Fig. <ref> with an isovalue interval [0, 0.2]. As there is no topological change happening in the evolution process, all Betti numbers β_0, β_1 and β_2 remain constant, and λ^T_1, λ^C_1 and λ^N_1 are continuous throughout the whole process. Both λ^C_1 and λ^N_1  decrease as the isovalue increases. 

Fig. <ref> illustrates the results for the kitten mode with one tunnel formed by its tail. The isovalue interval [0, 8] is considered. One can see all variables are continuous during the evolution process except that β_1 and λ^C_1 both drop at the same isovalue, where β_1 changes from 1 to 0. This happens due to the disappearance of the tunnel. In addition, λ^T_1 increases at the beginning, and then slows down its rate of increase at the isovalue after the tunnel disappears, and λ^C_1 and λ^N_1 decrease during the evolution process.

Note that there are also tunnels in the evolving manifolds for the genes-3 model, as we expected, a similar phenomenon can also be observed in Fig. <ref> for the change of the Betti numbers and the first non-zero eigenvalues. The isovalue interval [0.1, 4] is considered for this model. The disappearance of the three tunnels leads to a drop of β_1 from 3 to 0 and also a drop of λ^C_1. λ^T_1 initially increases, and then changes its behavior to decrease after the tunnels vanish. The evolution process results in a decrease in λ^C_1 and λ^N_1, just as the previous two models.

The evolving process of the four-ball model with isovalue interval [2, 3.84], see Fig.  <ref>, leads to discontinuities in all Betti numbers and the first non-zero eigenvalues. As the four separate components merge in the evolution, β_0 changes from 4 to 1, along with a drop in λ^T_1 at the same isovalue. In addition, β_1 increases from 0 to 3 due to the appearance of three tunnels when the merge happens and then decreases to 0 after the disappearance of all tunnels. The non-zero eigenvalue λ^C_1 has a drop that occurs when the tunnel vanishes, however, it is continuous when the tunnels are formed. This suggests that the continuity of λ^C_1 is only related to the death but not the birth of tunnels. One can also observe a slowdown in the rate of change of λ^T_1 following the disappearance of all tunnels. As the isolate increases further, a cavity occurs in the manifold, resulting in an increase of β_2 from 0 to 1 and finally a decrease from 1 to 0 after the cavity disappears. This topological change can also be observed in λ^N_1, where λ^N_1 becomes non-differentiable.



As illustrated by these models, changes in Betti numbers β_0, β_1 and β_2 and the first non-zero eigenvalues λ^T_1, λ^C_1 and λ^N_1 not only reflect the changes in topology, but also characterizes the changes in geometry for the evolution of manifolds. The rich information revealed by these variables leads to potential applications in various topological data analysis tasks. 








§ PROOF-OF-PRINCIPLE EXPERIMENTATION


In this section, we carry out a proof-of-principle experimental demonstration of the proposed persistent de Rham-Hodge theory. In this approach, the problem is defined on manifolds with boundaries. Appropriate boundary conditions are implemented to match actual topological dimensions. The resulting persistent Hodge Laplacians are solved to deliver the corresponding series of eigenvectors and eigenvalues at various scales. In this approach, we use these eigenvalues for machine learning predictions of protein-ligand binding affinity.  
The binding affinity describes the strength of protein-ligand interactions for each protein-ligand complex.

We consider two benchmark datasets, PDBbind-v2007 and PDBbind-v2016 <cit.>, to demonstrate the effectiveness of our framework in capturing the topological features of protein-ligand complexes. The datasets can be downloaded from http://pdbbind.org.cn/. These two PDBbind datasets provide collections of biomolecular complexes in Protein Data Bank (PDB) with experimentally a measured binding affinity for each protein-ligand complex, and are commonly used in various studies such as drug-discovery or molecular recognition, etc <cit.>. We aim to build a machine learning model, by utilizing the topological and geometric features of the protein-ligand complexes generated using our persistent Hodge Laplacian (PHL) framework as inputs, for predicting the protein-ligand binding affinities. 

The biomolecular complexes in each PDBbind dataset are organized into three sets, including a general set, a refined set and a core set, with each set being a superset of the next. In our experiments, for each dataset, we use the refined set, excluding the core set, to train the predictive model for the binding affinities of the protein-ligand complexes in the core set. The PDBbind-v2007 dataset contains a total of 1,300 complexes with 1,090 in the refined set and 195 in the core set, while the PDBbind-v2016 dataset has a total of 4,057 complexes with 3,767 in the refined set and 290 in the PDBbind core set.




 §.§ Element specific discrete to continuum mapping
 


The original datasets contain atomic names and coordinates, which are the so-called point cloud data. To generate manifold representations, we carry out the discrete to continuum mapping using the flexibility and rigidity index <cit.>. To compute the topological feature of each protein-ligand complex for the machine learning model, we use the element-specific approach <cit.>.  Specifically, we consider the pairwise interactions between element types that are commonly found in proteins and ligands, including Hydrogen (H), Carbon (C), Nitrogen (N), Oxygen (O), and Sulfur (S) in proteins, and Hydrogen (H), Carbon (C), Nitrogen (N), Oxygen (O), Sulfur (S), Phosphorus (P), Fluorine (F), Chlorine (Cl), Bromine (Br), and Iodine (I) in ligands. These interactions result in a total of 50 pairs of atom types for each protein-ligand complex <cit.>. However, due to the absence of H in most proteins, we reduce the number of atom pairs to 40 in practice, ignoring the element H in all proteins. These 40 atom pairs, formed by atom types {C, N, O, S} in proteins, and atom types {H, C, N, O, S, P, F, Cl, Br, I} in ligands, along with their xyz coordinates, are used to generate the topological features for each protein-ligand complex. In this paper, all atom-pair complexes are determined by a cutoff distance 12A from the ligand.













Let {𝐱^α_i, i=1,⋯, s} be the location coordinates of all s atoms in an atom pair, where α denotes the atom type of the atom either in the protein or in the ligand. For this atom pair, a level set function can then be obtained by considering the negative sum of Gaussian density functions defined at the xyz coordinates of all atoms, given as

    ρ(𝐱, τ) = -∑^s_i = 1exp(-(||𝐱 - 𝐱^α_𝐢||/τ r^α_i)^2),

where ||𝐱 - 𝐱^α_𝐢|| is the Euclidean distance from position 𝐱 to the location 𝐱^α_𝐢 of the i-th atom, τ is a scalar value, and r^α_i is the van der Waals radius of the i-th atom, determined by the atom type α. Given an isovalue c, the sublevel set

    M = {𝐱 | ρ(𝐱, τ)≤ c }

defines a compact manifold in ^3 with its boundary given by the isosurface ∂ M = {𝐱  |  ρ(𝐱, τ) = c }. A filtration of a manifold for the atom pair can then be obtained by choosing a list of evenly spaced isovalues of this level set function (<ref>). Let c_1< c_2<⋯ < c_s be such isovalues. We have their corresponding sublevel sets given as follows

    M_1⊂ M_2⊂⋯⊂ M_s,

where M_i is the compact manifold associated to isovalue c_i. In Fig. <ref> we present one example of the resulting filtration of manifolds at 3 different isovalues for atom pair OH in protein-ligand complex 4tmn. Note that the function (<ref>) is a special case of the flexibility rigidity index (FRI) density function <cit.>, which has been shown computationally stable in converting discrete point cloud representations to continuous embeddings, and been used for generating protein boundary surfaces <cit.> and interactive manifolds <cit.>. Therefore, one can also make other reasonable choices of FRI density functions to generate the filtration of manifolds. 




 §.§ Machine learning feature extraction



In the computation of the Laplacians, one can ideally choose a common Cartesian grid such that it contains all manifolds of interest for all protein-ligand complexes, which ensures that all Laplacians are computed consistently, making their spectra comparable for different complexes. However, as atoms are spread out in the space for different atom pairs, we need to use a sufficiently large grid with a fine resolution for accurate computation of Laplacians, which significantly increases the computational load. Instead, we consider, for each type of atom pairs, a fixed Cartesian grid, regardless of the types of protein-ligand complexes. This approach also ensures that the topological features are comparable for different protein-ligand complexes, as all spectra are computed in a same grid for all atom pairs of the same type. For simplicity, we choose a fixed grid spacing for all Cartesian grids across different atom pairs. 


We consider 9 evenly spaced isovalues in the interval [-0.5, -0.001] for all level set functions, which provide 9 compact manifolds for each atom pair. Note that the level set function (<ref>) is always less than 0 and approaches 0 as the norm of 𝐱 increases. This interval is chosen as isovalues greater than -0.001 result in no change on the 0-th Betti number β_0 of manifolds for most atom pairs, and isovalues smaller than -0.5 leads to high computational cost, as finer grids are necessary to resolve those isosurfaces. To ensure that the computation of Laplacians is accurate and no topological information is missing due to numerical errors caused by low resolution, we require that at least 8 grid cells of the Cartesian grid are contained in each connected component of a manifold. We compute, for each manifold, the BIG Laplacian L_3,n under the normal boundary condition, for which the number of its 0 eigenvalues gives the 0-th Betti number β_0. We then use the 0-th Betti number β_0 and the first k non-zero eigenvalues of L_3,n, as the topological feature for the manifold. These k+1 features for each of the 9 compact manifolds for each atom pair, amount to (k+1)×9×40 topological features for each protein-ligand complex. While we only used 9 isovalues within this interval for generating the manifolds in our experiments, more isovalues can be considered, which gives a filtration of more manifolds for each atom pair, and finally leads to more topological features for each protein-ligand complex. 


The spectra of the 0-th Laplacian, which in our case corresponds to L_3,n under the normal boundary condition, have proven effective and successful in many machine learning tasks <cit.>. While the Laplacians of other orders could also be used for generating more topological features, we utilize, in this preliminary test, only the spectra of L_3,n as features for the protein-ligand complexes in the machine learning model due to the computation efficiency. The results, as shown in Sec.<ref>, indicate that these features are sufficient to validate our framework in the machine learning task for predicting the protein-ligand binding affinities. 



 §.§ Machine learning algorithm


The machine learning models for predicting protein-ligand binding affinities often fall into two categories depending on the type of input data: complex-based or sequence-based models. The complex-based methods are trained using features obtained from the 3D protein-ligand complexes, while the sequence-based models learn from the one-dimensional protein sequences and the ligand simplified molecular-input line-entry system (SMILES) strings. In our experiments, besides the topological features from the 3D protein-ligand complexes, we incorporate protein-ligand features obtained from sequence-based models to build consensus models. To be specific, we make use of the recent pre-trained transformer protein language model Evolutionary Scale Modeling-2 (ESM-2) <cit.>, and the pre-trained Transformer-CPZ model <cit.> for generating the protein and ligand features, respectively, and use their concatenation as inputs for the binding affinity prediction. The residue embeddings from the last layer of the pre-trained ESM-2 model esm.pretrained.esm2_t33_650M_UR50D are used as the protein features, while the embeddings from the last layer of the pre-trained Transformer-CPZ model chembl27_pubchem_zinc_512 are used as the ligand features.


With the topological features and the embedding features obtained from ESM-2 and Transformer-CPZ, we employ the Gradient Boosting Regressor (GBR) module from Scikit-learn 1.4.2 for predicting the protein-ligand binding affinities. We then use the consensus prediction from these models as the final results. The GBR parameters used in our experiments are: n_estimators=10,000, max_depth=5, min_samples_split=5, learning_rate=0.005, loss=squared_error, subsample=0.5, and max_features=sqrt. Changing these parameters does not result in significant differences. To address the randomness from the machine learning algorithm, we repeat each modeling process 20 times with different random seeds, and use the average of predictive results. The Pearson correlation coefficients (PCC) are used as the evaluation metric to assess the performance of our proposed models.



 §.§ Experimental Results








The number of topological features for each protein-ligand complex, as in Sec. <ref>, is given by (1+k)×9×40, where k denotes the number of the first k non-zero eigenvalues of the Laplacians. To find the optimal parameter k leading to the best performance of predictive modules, we carry out the five-fold cross-validation on the training set of each PDBbind dataset with varying values of k based on the average of PCC values. The results suggest that k=5 provides the best PCC values for the PDBbind-v2007 training set, while k=10 gives the best PCC values for the PDBbind-v2016 training set. This results in a total of 2,160 topological features for each protein-ligand complex in the PDBbind-v2007 dataset and 3,960 topological features for each protein-ligand complex in the PDBbind-v2016 dataset. These topological features, along with the concatenated protein-ligand features from ESM-2 and Transformer-CPZ, are then used as inputs of the gradient-boosting regressor for binding affinity prediction. 

In Table <ref>, we report the average PCC values and the average root mean squared error (RMSE) of our models on the test set for each PDBbind dataset using only the topological features from PHL, the model using only the transformer features (TF), and the consensus module using both types of features. With the incorporation of topological features, one can see a significant improvement in PCC values when using the proposed consensus model for each dataset, compared to the model using only TF features. The best performance is achieved when using the consensus model, yielding a PCC value of 0.826 with RMSE given as 1.954 for PDBbind-v2007 and 0.849 with RMSE 1.728 for PDBbind-v2016. In addition, we present the Pearson correlation coefficients obtained from our model and those in the previous studies, with results from <cit.>. As illustrated in Fig. <ref>, our model outperforms all the other models for the two PDBbind datasets. These results demonstrate the utility and effectiveness of our method in capturing the topological features. 




§ CONCLUSION

Motivated by the success of TDA in machine learning, particularly, topological deep learning on point clouds <cit.>, we presented a new method, persistent Hodge Laplacian (PHL) in the Eulerian representation,  for generating topological and geometric characterizations of data on manifolds. Our approach differs from existing state-of-the-art TDA methods on point clouds in that the construction of PHLs is on a regular Cartesian grid for problems defined on manifolds. Instead of relying on simplicial complexes for building persistent combinatorial Laplacians, we offer two discrete Hodge stars that mimic the continuous operator and developed both a continuous theory for mapping of normal forms across manifolds in a filtration to enable persistent cohomology analysis and the associated discrete construction on structure-persevering Cartesian grids. A proof-of-principle test on two benchmark datasets validates our model and discretization, highlighting the simplicity and promise of the proposed  PHL machinery for machine learning and deep learning predictions of data on manifolds. 




§ ACKNOWLEDGMENTS

This work was supported in part by NIH grants R01GM126189, R01AI164266, and R35GM148196, National Science Foundation grants DMS2052983, DMS-1761320, and IIS-1900473, NASA  grant 80NSSC21M0023,   Michigan State University Research Foundation, and  Bristol-Myers Squibb  65109. The authors thank Dr. Hongsong Feng for his help on transformer embeddings.  






























































 Paper Title: Constraint maps: singularities vs free boundaries.

























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Constraint maps: singularities vs free boundaries]
Constraint maps: singularities vs free boundaries


 

A. Figalli]Alessio Figalli
alessio.figalli@math.ethz.ch
Department of Mathematics, ETH Zürich,  Raemistrasse 101, 8092 Zürich, Switzerland 

A. Guerra]André Guerra
andre.guerra@eth-its.ethz.ch
Institute for Theoretical Studies, ETH Zürich,  CLV, Clausiusstrasse 47, 8006 Zürich, Switzerland

S. Kim]Sunghan Kim
sunghan.kim@math.uu.se
Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

H. Shahgholian]Henrik Shahgholian
henriksh@kth.se
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden





§ ABSTRACT 
Energy-minimizing constraint maps are a natural extension of the obstacle problem within a vectorial framework. Due to inherent topological constraints, these maps manifest a diverse structure that includes singularities similar to harmonic maps, branch points reminiscent of minimal surfaces, and the intricate free-boundary behavior of the obstacle problem. 
The complexity of these maps poses significant challenges to their analysis.


In this paper, we first focus on constraint maps with uniformly convex obstacles and establish continuity (and therefore higher-order regularity) within a uniform neighborhood of the free boundary. More precisely, thanks to a new quantitative unique continuation principle near singularities (which is new even in the setting of classical harmonic maps), we prove that, in the uniformly convex setting, topological singularities can only lie in the interior of the contact set. We then establish the optimality of this result.

Second, while exploring the structure of the free boundary, we investigate the presence of branch points and show how they lead to completely new types of singularities not present in the scalar case.






[
    [
    August 5, 2024
==================









§ INTRODUCTION



Let M be a smooth domain with compact, connected complement in ^m, m ≥ 2, and let Ω be a bounded domain in ^n, with n≥ 1. In this paper we consider maps u∈ W^1,2(Ω;M) minimizing the Dirichlet energy

    [v] := ∫_Ω |Dv|^2 dx,

among all admissible maps v∈ W^1,2(Ω;M) with v - u ∈ W_0^1,2(Ω;^m). To emphasize the target constraint u ∈M as well as its energy minimality, we call such maps minimizing constraint maps. As shown in <cit.>, minimizing constraint maps are weak solutions to the Euler–Lagrange system

    Δ u = A_u(Du,Du)χ_u^-1(∂ M)  in Ω,

where A is the second fundamental form of the boundary ∂ M relative to M.  

We shall call M^c := ^m∖ M the obstacle, as the image of the constraint maps is not allowed to go inside it. The presence of the obstacle naturally induces the free boundary[The notion of free boundary here differs significantly from the one in the theories of harmonic maps and minimal surfaces, as our notion resides in the domain, while the latter appears in the target.]  Ω∩∂ u^-1(M), which is the interface between the non-coincidence set u^-1(M) and the coincidence set u^-1(∂ M). System (<ref>) shows that, when restricted to int(u^-1(∂ M))⊂Ω, constraint maps are harmonic maps into ∂ M. At the same time, each component of a constraint map is harmonic in int(u^-1(M)). This variational problem hence can be seen as a canonical extension of the scalar obstacle problem into the vectorial setting. 

Being vectorial maps, constraint maps develop in general discontinuities[Although the term “singularities” is more common in the Harmonic Maps literature, here we have chosen to use the word “discontinuities” to ensure that the Free Boundary community does not confuse them with singularities of the free boundaries.] 
due to topological differences between the domain and their image in the target.  Their singular set is denoted   throughout the paper by 

    Σ(u) := { x∈Ω: u is not continuous at x}.

Besides discontinuities, constraint maps can also exhibit branching, which is concerned with the regularity of the image of the map and is encapsulated by the branch set

    (u) := ⋃_k=0^min{n,m}-1_k(u),    _k(u) := {x∈Ω∖Σ(u): rank(Du(x))=k }.

Here, the word branching is motivated by the connection with minimal surfaces and geometric function theory,  e.g.  the map  u z↦ (z^2,  z^ℓ/2), defined for z∈=^2, is such that (u)= _0(u)= {0}. Points in (u) can either be true branch points,
such as when ℓ is odd, or false branch points, 
such as when ℓ is even.
More precisely, false branch points correspond to a degeneracy of the parametrization and not of the actual image that is being parametrized, while true branch points correspond to singularities of the image; see Figure <ref> for an illustration. We will be especially concerned with the case k=0 since the right-hand side in (<ref>) vanishes at points in _0(u), making the obstacle problem degenerate. 

 


Interestingly, constraint maps reveal an extremely rare interplay between ∂ u^-1(M), Σ(u), and (u) altogether. Despite the long history of constraint maps since their initial appearance in the literature, dating back to the 1970s by the work of Hilderbrandt <cit.> and Tomi <cit.>, the interaction between these apparently correlated objects was addressed only recently by the authors <cit.> in a different but related setting. 
The goal of this paper is to address the following fundamental question: 


    
    Does the free boundary meet the mapping singularities manifested by Σ(u) or (u)?
    
    
    
    
    


This is an inherently difficult question,  
as answering it requires an understanding of the interplay between local analytic aspects and global geometric and topological  constraints.



One of the major findings of this paper is that there are no discontinuities of the maps near their free boundaries, provided that the obstacle is uniformly convex[Uniform convexity of M^c means that there exists R>0 such that the following holds: for any point x ∈∂ M there exists a ball B_R(y)⊃ M^c such that x ∈∂ B_R(y).].
This condition is essentially optimal: we find a large class of general convex obstacles which produce discontinuities on the free boundaries. 
To our knowledge, and despite the significance of this problem, the first suggestion of a possible resolution of discontinuities by free boundaries was raised in the authors’ recent work <cit.> in a different but related setting. 

The second major finding of this paper is that, even when the map is regular and the obstacle is a ball, the map may exhibit branching
on the free boundary in such a way that the free boundary itself becomes singular. Indeed, it is  important to observe that the right-hand side in (<ref>) becomes zero at all points within _0(u). Consequently, at these points, the behavior of the free boundary may exhibit significant irregularities, which do not occur in the classical scalar obstacle problem.




As we shall explain later, once discontinuities and branch points are understood,  the free boundary regularity can be reduced to that for the scalar obstacle problem (cf. <cit.>).






 §.§ A brief overview of the literature


The study of minimizing constraint maps as solutions to vectorial obstacle problems dates back to the 1970s. Pioneering work by Hildebrandt <cit.>, and Tomi <cit.> independently addressed the regularity problems for constraint maps of two independent variables, employing different methodologies.
Early advances in higher dimensions can be attributed to the work of Hildebrandt-Widman <cit.>, who considered convex target constraints. The first comprehensive result in higher dimensions was established by Duzaar <cit.>, who proved an ε-regularity theorem. As a consequence, a partial regularity theorem was derived, affirming the W^2,p-regularity of mappings outside a closed singular set with dimension less than n-2. Subsequently, Duzaar–Fuchs <cit.> proved that the singular set has dimension at most n-3. Although this result is optimal in general, it was later observed in <cit.> that, under certain topological and geometric conditions on the target constraint, discontinuities are not present.
We note that the development of these works closely followed the higher-dimensional theory of minimizing harmonic maps, pioneered by Schoen–Uhlenbeck <cit.>: as for harmonic maps, 
the theory of constraint maps has been generalized to the setting of the p-Dirichlet energy in <cit.>, almost minimizing maps in <cit.>,  and weakly constraint maps as well as the associated heat flow problem in <cit.>. The regularity results in two space dimensions were recently extended to weakly constraint maps in <cit.>. We also refer the reader to <cit.> for results in the case where the obstacle is thin, in the sense that it has higher codimension.

It is worth noting that, in the simple case where the domain is one-dimensional, minimizing constraint maps are simply energy-minimizing geodesics on the manifold-with-boundary M. The regularity and (non-)uniqueness of geodesics on manifolds-with-boundary were studied in depth in <cit.> and also more recently in <cit.>. 

Quite surprisingly, and despite all of the above results, the theory did not progress much since the 1990s, until recently three of the authors revived the problem in <cit.> from the perspective of free boundary problems, opening up new and delicate issues on the exact regularity of the maps away from their singular set and around their free boundaries. Although the study of minimizing constraint maps had its origin as a vectorial obstacle problem, historically only basic properties of the free boundaries were shown in <cit.>. The result closest to this paper is, in fact, the very recent work <cit.> of ours concerning constraint maps minimizing a vectorial Alt–Caffarelli energy,  which corresponds to a vectorial version of the Bernoulli problem.

In recent years there has been a surge of interest from the free boundary community towards extending scalar problems into vectorial settings, see for instance <cit.> for Bernoulli-type problems from diverse angles,  <cit.> for obstacle-type problems,  and <cit.> for thin-obstacle-type problems. All the aforementioned works successfully extend aspects of the scalar theory and explore inherently interesting issues related to the branching of free boundaries from several components. 

However, our minimizing constraint maps exhibit very different characteristics compared to the solutions to the aforementioned vectorial problems, due to the possible development of discontinuities of the map itself around the free boundaries. Hence a complete understanding of constraint maps requires bridging the theory of harmonic maps with that of free boundary problems. 





 §.§ Main results


Our first result asserts that, whenever the obstacle is uniformly convex, minimizing constraint maps are uniformly C^1,1-regular in a universal neighborhood of the non-coincidence set (hence, in a set which strictly includes the free boundary). 
It should be noted that C^1,1-regularity is optimal, due to the classical theory of the obstacle problem. 
In addition, the result cannot be improved: solutions cannot be continuous everywhere neither in the interior of the coincidence set,  due to the possible presence of singularities coming from minimizing harmonic maps with values in ∂ M.


Let M⊂^m be a smooth domain with uniformly convex complement. There exist constants δ∈ (0,1/2) and c > 1, depending only on n, m, ∂ M, and Λ, such that for any minimizing constraint map u∈ W^1,2(B_2;M) satisfying u _W^1,2(B_2)≤Λ, we have u ∈ C^1,1(B_1∩ B_δ(u^-1(M))) with the estimate 

    u _C^1,1(B_1∩ B_δ(u^-1(M)))≤ c.



To describe the strategy and challenges in the proof of Theorem <ref>, it is fruitful to introduce the distance function to ∂ M, which we denote by ρ.
Note that the smoothness and convexity of M^c guarantee that 
ρ is smooth in M. Then (<ref>) 
yields an Euler–Lagrange equation for ρ∘ u, namely

    Δ(ρ∘ u )     = ρ_u(Du,Du) χ_u^-1(M),

where ρ_u denotes the Hessian of ρ evaluated at u. Noting that u^-1(M) is precisely the positivity set of ρ∘ u, we observe that (<ref>) is a scalar obstacle-type equation, but with an important caveat. Indeed, the nonlinearity ρ_u(Du,Du), which acts as a force in (<ref>), 
can become unbounded (when u is discontinuous) and can also vanish (when either ρ_u, Du, or their bilinear combination vanishes). Hence ρ∘ u can behave quite differently from a solution to the scalar obstacle problem, where one deals with forcing terms that are regular and bounded away from zero. In particular, ρ∘ u has no inherent non-degeneracy: as we show in Theorem <ref> below, it may in fact vanish to arbitrarily high order on the free boundary.

In our previous work <cit.> (in a slightly different context) we observed that the distance map ρ∘ u is continuous everywhere, even across Σ(u), and that the non-coincidence set u^-1(M) has null density at any point of Σ(u): at any (small) scale r, we have 

    |B_r(x_0)∩ u^-1(M)| = o(r^n),  ∀ x_0∈Σ(u).

Interestingly, the assertion (<ref>) is proved via the unique continuation principle (UCP) for constraint/harmonic maps. However, to show that Σ(u) is completely disjoint from the free boundary, we need the much stronger assertion that ρ∘ u  vanishes identically in a full neighborhood of Σ(u), i.e. there is some small r_0>0 such that 

    B_r_0(x_0)⊂ u^-1(∂ M),  ∀ x_0∈Σ(u).
 
Our idea is then as follows. We look at the distance map ρ∘ u, which is supported exactly on u^-1(M), and is subharmonic for convex obstacles M^c. A key property of ρ∘ u is that it vanishes continuously at any point of Σ(u) whence, upon translating and rescaling, we may assume that 0≤ρ∘ u ≤ in B_2 and 0∈Σ(u), for a given > 0. Then, by a lemma reminiscent of the De Giorgi argument (Lemma <ref>), we observe that ρ∘ u = 0 in B_1, provided that  is sufficiently small and  

    |B_r∩ u^-1(M)|≤c/(R-r)^βρ∘ u_L^∞(B_R)^α,  ∀  1 ≤ r< R ≤ 2,

for some α, β, and c that do not depend on the scales r and R. Condition (<ref>) can be considered as the reverse of the local maximum principle for subharmonic functions. Since we are free to choose the smallness parameter  for the sup-norm of ρ∘ u over B_2, our task boils down to verifying (<ref>). 

What appears to be intriguing is the involvement in (<ref>) of the vectorial character of u implied by the presence of 0∈Σ(u) (note that u^-1(M) is simply the set where ρ∘ u is positive). To exploit the hidden vectorial character in (<ref>), we resort to the Euler-Lagrange equation (<ref>) for ρ∘ u. Integrating both sides of (<ref>), and invoking the uniform convexity of M^c, we obtain around 0∈Σ(u),[That 0∈Σ(u) is needed to get ρ_u(Du,Du)≈ |Du|^2 in the L^1-sense. In general, we always have ρ_u(Du,Du)≈ |D(Π∘ u)|^2 a.e., with Π being the nearest point projection to ∂ M (by (<ref>) and (<ref>)). However, ρ∘ u becomes negligible around any discontinuity point of u, so we obtain the last inequality in (<ref>).] along a chain of elementary tools (see (<ref>) – (<ref>)), that 

    1/(R-r)^2∫_B_R (ρ∘ u) dx ≳∫_B_r∩ u^-1(M)ρ_u(Du,Du)  dx ≳∫_B_r∩ u^-1(M) |Du|^2  dx

for every 1≤ r<R≤ 2. 

Our analysis would have been softer if |Du| ≳ 1 uniformly in B_r, as the rightmost side of (<ref>) would then readily yield (<ref>) with w = ρ∘ u. Since |Du| blows up at 0∈Σ(u) possibly with a rate 1/r in B_r, |Du|≳ 1 is expected to be true at least for a large portion of the neighborhood of 0∈Σ(u).
However, |Du| may not behave so uniformly near its singularity. In principle, |Du| may become very small in a set of tiny measure (and even vanish somewhere inside), and such a tiny set may just happen to be the non-coincidence set u^-1(M) as we are close to Σ(u) (recall (<ref>)): e.g., u^-1(M) is a cusp with its tip on Σ(u), such that one of the tangent maps ϕ of u is of rank zero along the direction of the cusp. The latter would certainly be the worst case scenario for us, as the rightmost side of (<ref>) sees u^-1(M) only. 

Thus, our analysis calls for a more robust approach that works in the absence of pointwise control of the vanishing behavior of |Du|. For this purpose, our new key result is an L^-γ-estimate of |Du| near Σ(u). Note that the next statement does not require convexity of the obstacle.



  
 
     Let M⊂^m be a smooth domain with compact complement and let u∈ W^1,2(B_4;M) be a minimizing constraint map satisfying u _W^1,2(B_4)≤Λ. There exist _0, γ > 0, both depending only on n, m and ∂ M, and a constant c>1, depending further on Λ, such that if ∫_B_1(y) |Du|^2 dx ≥_0^2 for some y∈ B_2 then 

    ∫_B_1 |Du|^-γ dx ≤ c.

 

 

This estimate can be understood as a quantitative version of the UCP for the constraint maps, as it shows that |Du| cannot vanish too fast around Σ(u). Once such an estimate is available, one can use the generalized Hölder inequality to verify the reverse link (<ref>) (see (<ref>)), which in turn yields (<ref>) as desired. Therefore, Theorem <ref> lies at the heart of our analysis for Theorem <ref>.

 
As already observed above, Theorem <ref> works for general smooth compact obstacles M^c without any condition on their principal curvatures. Moreover, this theorem also holds for minimizing harmonic maps into smooth compact manifolds without boundary, and it is even new in the latter context. 


 


    The above results of UCP-type are somewhat reminiscent of the classical quantization results for minimizing harmonic maps <cit.>, which assert  that there is a universal lower bound on the energy of smooth harmonic maps which are not homotopic to a constant.  The theorem  is also evocative of a result from <cit.>, where the authors prove an ^n-2-measure estimate on the set _0(u) for a minimizing harmonic map u; however,  this estimate is qualitatively different from (<ref>), and neither of them implies the other. 

    
    
    We emphasize that the structure of blow-ups of harmonic maps near their discontinuity points is only well-understood in some very specific instances, such as when n=3 and N = 𝕊^2 <cit.>,  or n=4 and N = 𝕊 ^3 <cit.>. In these cases, the structure of the blow-ups can be exploited to prove (<ref>), as we will see in Section <ref>. However, since blow-ups are poorly understood in general, we are forced to rely on a fundamentally different type of argument to prove Theorem <ref>. 


    The key observation for proving Theorem <ref> is the existence of a critical scale at every regular point such that,
    at this scale, we can control both the regularity and the frequency of |Du| uniformly and in our favor. The critical scale plays a  significant role in two ways. Above this scale, the effect of the discontinuous singularity dominates, so that |Du|≳ 1 in a large portion of the neighborhoods. Below the critical scale we have |Du| ≪ 1, but we can still control the doubling property of |Du| in a uniform way. This is because the regularity of u is uniformly controlled on such a scale. In other words, 1/|Du| does not grow so fast in a large portion of the neighborhoods on these scales. Finally, we can patch these two different scales altogether via a Calderón–Zygmund decomposition argument, which leads us to the desired L^-γ-estimate for |Du|. 
    
    Let us address that the notion of the critical scale here is closely related to that of the regularity scale introduced in <cit.> in the study of the rectifiability properties of Σ(u).  However, the analysis in these works is different from ours,  as the authors there are concerned with obtaining (optimal) L^p-estimates for |Du| with p > 1.


      


Returning to Theorem <ref>, let us now address the optimality of the uniform convexity condition. Our next result asserts that this condition is essentially optimal,  as the conclusion of Theorem <ref> no longer holds as soon as ∂ M contains a portion of a hyperplane, even when M^c is convex.   We remark that by <cit.>, the convexity of M^c ensures a well-defined free boundary even in the presence of discontinuities, as it implies that (u,∂ M) is continuous.  In fact,  we can show that for obstacles with flat sides, minimizing constraint maps whose boundary values have non-trivial topological degree (see Appendix <ref> for the definition) always develop discontinuities on their free boundaries:




Let Ω⊂^n be a  smooth uniformly convex  domain, and let M⊂^n be a smooth domain with convex complement such that  M^c⋐Ω and ∂ M contains a relatively open subset T of a hyperplane. Let n≥ 3 and  consider boundary data g∈ C^∞(∂Ω;∂Ω) with (g) ≠ 0. If u ∈ W_g^1,2(Ω;M) = { u ∈ W^1,2(Ω;M);  u = g  on ∂Ω} 
is a minimizing constraint map, then 

    ∂ u^-1(M)∩Σ(u) ≠∅.

In fact,   for every y∈ T there is {x_k}⊂Ω∩ u^-1(M)∖Σ(u) such that x_k→ x_0 ∈∂ u^-1(M)∩Σ(u) and u(x_k)→ y, and so we write

    T⊆ u( ∂ u^-1(M)∩Σ(u)).
  



The assumption n≥ 3 in Theorem <ref> is used only to guarantee the existence of a minimizing constraint map. See Remark <ref> below for further discussions.


The situation for non-convex obstacles is even more exotic than the one described above: as a byproduct of Theorem <ref>,  we show in Corollary <ref> that, for such obstacles,  the non-coincidence set u^-1(M) may not be open.[It is not immediately clear how to define u^-1(M) in general, since u may develop discontinuities. A robust definition, which is independent of the choice of representative for u∈ W^1,2(Ω;M), is to set 

    u^-1(M):={x∈Ω: lim inf_r→ 0ρ∘ u_L^∞(B_r(x))>0}.

This set contains those points in Ω∖Σ(u) which are mapped by (the smooth representative of) u|_Ω∖Σ(u) to M.]
This yields in particular the sharpness of <cit.>.


Producing examples as those in Theorem <ref> is challenging, since we do not know how to construct explicit minimizing maps  in non-perturbative regimes. Indeed,  due to the vectorial nature of the problem, minimizers generally do not inherit the symmetry of their boundary data; see e.g. <cit.> for a dramatic symmetry-breaking phenomenon in the classical case of harmonic maps from the unit ball 𝔹^3 to 𝕊^2.  In particular, our approach to Theorem <ref> is based on a topological degree argument and is necessarily global, in contrast to the typical local analysis performed in the context of free boundary problems.





The above results give a rather complete picture of the behavior of minimizing constraint maps near their free boundaries but do not address the structure of the free boundary itself.  The main difficulty in the analysis is due to the possible occurrence of branch points on the free boundary,  and in particular of points in _0(u): at such points, no matter what the obstacle is, the right-hand side in (<ref>) vanishes and so we are faced with a degenerate obstacle problem. On the other hand, if the obstacle is uniformly convex, then by Theorem <ref> and the classical theory for the obstacle problem, the structure of the free boundary is well-understood in ∂ u^-1(M)∖_0(u), see <cit.>. 
We thus arrive at the following questions: For uniformly convex obstacles, is the set ∂ u^-1(M)∩_0(u) non-empty? If so, how does the free boundary look like around such points?




The structure of the free boundary for degenerate free boundary problems is a difficult topic. In this direction, very little is known even in the scalar setting, except e.g. in the particular examples studied in <cit.>. However, the degeneracy in these references stems from an absorbing term acting as an additional force in the problem; see also the discussions in <cit.> for a forcing term given directly by the obstacle. Nevertheless, the potential appearance of branch points for vectorial maps is part of the intrinsic nature of the problem and is not dictated by external forces. 

Here we construct and analyze a large class of examples of mininimizing constraint maps u∈ W^1,2(Ω;^3\ B^3) with non-trivial free boundaries such that

    ∂ u^-1(^3\ B^3)∩_0(u)≠∅,

thus showing that branch points can occur on free boundaries, even when the obstacle is a ball. 
In order to construct such examples, we shall consider maps that minimize the Dirichlet energy in the more restricted class of k-axially symmetric maps;
see Definition <ref>. Briefly, a map is called k-axially symmetric if it maps each cylinder centered on the z-axis to another such cylinder while rotating it k-times. Our analysis shows that, for k≥ 2, each free boundary point on the z-axis is simultaneously a branch point and a singular free boundary point: thus, branching can occur and can be responsible for free boundary singularities.







    Fix  k≥ 2, λ > 1, and a k-axially symmetric map g∈ C^2(^2;^2) with |(g)| = k. Let u∈ W_λ g^1,2( B^3;^3\ B^3) be an energy minimizer among k-axially symmetric maps. For any x_0∈∂{|u|>1}, there is a ball B⊂ B^3 centered at x_0 such that u∈ C^1,1(B) is the unique minimizing constraint map in W^1,2_u(B;^3\ B^3). Moreover, the following hold for the set _0(u) of rank zero points and the free boundary ∂{|u|>1}:
    
        
        
  *  _0(u)∩∂{|u|>1}∩∂( int{|u|=1})≠∅;
        
  *  |Du| vanishes exactly with order k-1 at every point of _0(u);
        
  *  ∂{|u|>1} cannot be represented as a C^1-graph in any neighborhood of _0(u) on the z-axis.
    
    In particular, (<ref>) and (<ref>) show that the free boundary is non-trivial and singular.


We refer the reader to Section <ref>, and in particular to Theorem <ref> therein, for a much more precise description of the structure of the free boundary near points in _0(u). As we shall see, in the setting above we can give a complete classification of the possible blow-ups appearing in the free boundary analysis.














 §.§ Organization of the paper


The paper is organized as follows. In Section <ref>, we describe the setting and notation of this paper and collect preliminary results on minimizing constraint maps. In Section <ref>, in preparation for our first main result, we prove the uniform continuity of the distance of a minimizing constraint map to the convex hull of the obstacle. Section <ref> is devoted to the proof of Theorem <ref>. In Section <ref> we prove Theorem <ref>. In Section <ref>, we study minimizing constraint maps for non-uniformly convex obstacles, and in particular we prove Theorem <ref>. 
Section <ref> collects some remarks and examples concerning branch points and their interaction with free boundaries. In Section <ref> we develop the regularity theory of axially symmetric constraint maps, and in particular we prove analogues of Theorems <ref> and <ref>  in this setting. Finally, in Section <ref> we study branch points on free boundaries for axially symmetric maps and prove Theorem <ref>. 

The paper also contains four appendices consisting of auxiliary results that are either known or follow with some relatively mild modifications from standard arguments. In Appendix <ref> we recall the definition and some basic properties of the topological degree, and its local variant. In Appendix <ref> we recall standard facts about the Almgren frequency of a solution to a regular linear elliptic system. In Appendix <ref> we prove a version of Luckhaus' lemma in the setting of axially symmetric constraint maps. Finally, in Appendix <ref> we prove the uniqueness and regularity of weakly constraint maps into small geodesic balls.









 §.§ Notation


For the reader's convenience, we gather here a list of notation to be used throughout this paper.


[l]l l 
    Ω        a bounded domain in ^n (n≥ 1)  
 
    Ω_η        { x ∈Ω:(x,∂Ω)>η}  (η≥ 0)    
 
    M         smooth domain in ^m (m≥ 2)  
 
    u         (u^1, ⋯ , u^m)    

    W^1,p(Ω;M)       {u∈ W^1,p(Ω;^m): u∈M a.e. in Ω}      

    W^1,p_g(Ω;M)       the Dirichlet class W^1,p(Ω;M)∩ (g+W^1,p_0(Ω;^m)), for g∈ W^1,p(Ω;M)   

    B^m         the open unit ball in ^m  
  
    S^m-1         the unit sphere in ^m, S^m=∂ B^m 
 
    A_p (· , ·)         the second fundamental form of ∂ M at p, see (<ref>)   
   
    (∂ M)         tubular neighborhood  of ∂ M 
    
   
    ρ         signed distance function to ∂ M, smooth on (∂ M)    
   
    ν=∇ρ         inward unit normal to M, extended to (∂ M) 
   
    Π          nearest point projection: (∂ M)→∂ M
       
 
    ν⊗ν          the tensor product matrix with entries (ν^iν^j)_ij 
   
    ξ^⊤         (I - ν⊗ν)ξ,    orthogonal projection of ξ into T(∂ M), see (<ref>)      
 
    [u]         ∫_Ω |Du|^2   dx   
 
    E(u,x_0,r)         r^2-n∫_B_r(x_0) |Du|^2 dx,  normalized energy     

    N(f,x_0,r)         r∫_B_r(x_0) |Df|^2 dx/∫_∂ B_r(x_0) |f - (f)_x_0,r|^2, the frequency function     

    Σ(u)          { x∈Ω: u is not continuous at x}    
   
    _k(u)          {x∈Ω∖Σ(u):Du has rank k at x}  
 
    ^k         k-dimensional Hausdorff measure    
   
    dim_         Hausdorff dimension     
    
    F:G         f_α^i g_α^i,  where  F = (f_α^i) and  G= (g_α^i) are matrices in ^m× n    
 
    D^k        the k-th order differential operator in the ambient space, D_α := ∂/∂ x_α 

    ∇^k        the k-th order differential operator in the target space, ∂_i := ∂/∂ y^i 

    f_y(ξ,ζ)         ∂_ij f(y)ξ_α^iζ_α^j, for a function f from the target space, and ξ,ζ∈^m× n 
 








§ PRELIMINARIES
 





 §.§ Problem setting


Throughout this paper, Ω⊂^n is a bounded domain and M⊂^m is a smooth domain; we always assume that n,m≥ 2. The  complement of M, M^c, is called the obstacle, and we always assume that it is compact. We denote by ρ the (signed) distance function to ∂ M that takes positive values in M, and by Π:^m →∂ M the nearest point projection onto ∂ M. We also write (∂ M) for the largest open neighborhood of ∂ M where ρ and Π are smooth; in particular, in the case where M^c is convex, the set (∂ M) strictly contains M.  We set ν:=∇ρ on (∂ M) and note that, on ∂ M, ν is the inward-pointing unit normal to ∂ M. See Figure <ref> for an illustration.




We denote by B_r(x_0) the ball of radius r centered at the point x_0, and more generally B_r(F) denotes the neighborhood of width r around the set F.  If B=B_r(x_0) is a ball, we write λ B := B_λ r(x_0)
for the dilation of B by a factor λ>0 around its center; we use a similar notation for cubes. We also denote by _(S) the Hausdorff dimension of a set S, by ^k(S) the k-dimensional Hausdorff measure of S, and by |S| the Lebesgue measure of S. We write D and ∇ for the usual differential operators in the ambient space and target spaces, respectively.  
Also, for a function f, we write f_y(ξ,ζ):= ∂_ij f(y)ξ_α^iζ_α^j.






 §.§ Known results for minimizing constraint maps


To keep our exposition as self-contained as possible,  we gather here some basic results about minimizing constraint maps. These can be seen as generalizations of classical theorems concerning minimizing harmonic maps into closed manifolds <cit.>. 

A basic quantity in the study of minimizing constraint maps is the normalized energy

    E(u,x_0,r) := r^2-n∫_B_r(x_0) |Du|^2 dx.

Minimizing constraint maps are stationary upon inner variations, and this yields a monotonicity formula for the normalized energy. A more general result is proved in <cit.>; here, we only present the statement adapted to the current setting. 


Let u∈ W^1,2(Ω;M) be a minimizing constraint map. For all x_0∈Ω we have

    E (u, x_0,s) - E (u, x_0,r) ≥ 2 ∫_B_s(x_0)∖ B_r(x_0) |x-x_0|^2-n|∂ u/∂ N|^2 dx 
        ∀   0<r<s<(x_0,∂Ω),

where ∂/∂ N denotes the directional derivative in the radial direction. In particular, if E(u,x_0,s) = E(u,x_0,r) then u is 0-homogeneous around x_0 inside B_s(x_0)∖ B_r(x_0), i.e., 

    (x-x_0) · Du = 0   a.e. in  B_s(x_0)∖ B_r(x_0).

  



A crucial tool in our analysis is the compactness of minimizing constraint maps, originally established in <cit.> within a broader context. Since we will frequently reference it here, we provide the adapted statement for the reader's convenience.


Let {u_k}_k=1^∞⊂ W^1,2(Ω;M) be a bounded sequence of minimizing constraint maps. Then there exists a minimizing constraint map u ∈ W^1,2(Ω;M) such that, up to a subsequence, u_k → u strongly in W_^1,2(Ω;^m). 



The monotonicity of the normalized energy yields a well-defined limit E(u,x_0,0^+) for every x_0∈Ω. 
As an immediate consequence of Lemmas <ref> and <ref>, we obtain the upper semicontinuity of the energy density. 


Let {u_k}_k=1^∞⊂ W^1,2(Ω;M) be a sequence of minimizing constraint maps such that u_k→ u strongly in W^1,2(Ω;^m). Then for any sequence {x_k}_k=1^∞⊂Ω with x_k → x∈Ω we have 

    E(u,x,0^+) ≥lim sup_k→∞ E(u_k,x_k,0^+).



We will also make use of the -regularity theorem, which states that minimizing constraint maps are regular in the vicinity of points with small energy. The optimal version of this result is due to the recent work <cit.> (see also <cit.> for a previous version of this statement).


 
There exist a small constant _0 > 0 
depending only on n, m, and ∂ M, such that the following holds: if u∈ W^1,2(Ω;M) is a minimizing constraint map with
 
    E(u,x_0,2r) ≤_0^2,    where  B_2r(x_0)⋐Ω,
 
 then u∈ C^1,1(B_r(x_0)), with the estimate  
 
    D^j u_L^∞(B_r(x_0))≤1/r^j,   j=1,2.

 

Recall from (<ref>) that Σ(u) consists of all discontinuity points of the map u in the prescribed domain. A useful corollary to Theorem <ref> is the energy quantization at points of Σ(u):
 


There exists a constant _0 > 0, depending only on n, m, and ∂ M, such that for any minimizing constraint map u∈ W^1,2(Ω;M) we have that x_0∈Σ(u) if and only if E(u,x_0,0^+) ≥_0^2. 


Through the usual dimension-reduction principle,  the above results yield an optimal partial regularity theorem, see <cit.>.





Let u∈ W^1,2(Ω;M) be a minimizing constraint map. Then u∈ C_^1,1(Ω∖Σ(u)) and _(Σ(u))≤ n-3.






The following lemma can be found in <cit.>, and will be used frequently in our subsequent analysis.




Let u∈ W^1,2(Ω;M) be a minimizing constraint map such that u(B)⊂(∂ M) a.e. in a ball B⊂Ω. Then 
    
    Δ (ρ∘ u) =  ρ_u(Du,Du)χ_u^-1(M)  in B,

    in the weak sense. 






 §.§ Vector calculus identities and geometric bounds


In this subsection we prove some elementary identities and estimates related to the geometry of ∂ M. 
Since ν = ν∘Π in (M), we have the decomposition 

    𝕀 = Π+ ρν = Π+ρ(ν∘Π).

In particular, differentiating both sides of this identity, we have - ν⊗ν = ( + ρ∇ν)∇Π; note that -ν⊗ν is the orthogonal projection onto the tangent bundle T(∂ M).
For a vector ξ∈^m, we thus write
    
    ξ^⊤ := ξ - (ν·ξ)ν= [( + ρ∇ν)∇Π]ξ

 for the orthogonal projections of ξ into T(∂ M).
 
The second fundamental form of ∂ M, appearing in (<ref>), can be expressed at points in ∂ M as


    ρ(ξ,ξ) = ρ(ξ^⊤,ξ^⊤) = -ν· A(ξ^⊤,ξ^⊤).

When the obstacle M^c is uniformly convex, we obtain a simple relation between ρ and the differential |∇Π|:





Let M be a smooth domain with uniformly convex complement. Then 

    ρ(ξ,ξ) ≥κ/1+κρ |(∇Π)ξ|^2  ∀ ξ∈^m,

where κ is the curvature lower bound for ∂ M. 







As M^c is convex, ρ, ν, and Π are well-defined and smooth in a neighborhood of M. Let {τ^1,⋯,τ^m-1,ν} be the principal coordinate system of ∂ M, and κ_i be the i-th principal curvature. As ν points outward to M^c, it follows from <cit.> that 

    ρ(τ^i,τ^i) = κ_i∘Π/1 + (κ_i∘Π)ρ  in M.

Also, as M^c is uniformly convex, κ_i∘Π≥κ in M, for some κ>0 which depends only. Since the map t↦t/1+ at is strictly increasing in t>0, for each a ≥ 0, we deduce from (<ref>) that

    ρ(ξ,ξ)≥κ/1+κρ |ξ^⊤|^2  in M  ∀ ξ∈^m.
 
Since ∇ν = ρ (as a consequence of the relation ν = ∇ρ), using again (<ref>) we see that

    ( + ρ∇ν)(τ^i,τ^i) = 1 + 2(κ_i∘Π)ρ/1+ (κ_i∘Π)ρ∈ [1,2]  in M.

In particular, combining (<ref>) with (<ref>) yields 

    |ξ^⊤|^2≥  |(∇Π)ξ|^2  in M  ∀ ξ∈^m.

Thus, (<ref>)  follows from (<ref>) and (<ref>). 










 §.§ Existence and examples of constraint maps


Consider boundary data g∈ C^∞(∂Ω; M). In this setting, the direct method yields existence of minimizing constraint maps in W^1,2_g(Ω;M), provided this space is non-empty. 
We discuss whether this holds in the next remark and for simplicity we take Ω= B^n to be the open unit ball.


In studying W^1,2_g( B^n;M), there are two different cases to consider.
    
    
  * If n≥ 3, then W_g^1,2( B^n;M)≠∅. Indeed, it suffices to consider the 0-homogeneous extension x↦ g(x/|x|) to obtain a map in that class. The crucial calculation is that the map x↦ x/|x| is in W^1,p near the origin if and only if n>p. Thus, in order to be able to take p=2, we require that n≥ 3.
    
  * If n=2, the situation is more subtle and it can be concisely expressed in terms of the homotopy extension property: W^1,2_g( B^2;M)≠∅ if and only if
    
    g  is homotopic to  G|_ S^1 for some continuous map  G B^2→M.

    Note that this characterization holds for boundary data g that are not necessarily smooth, but are just in the natural trace space H^1/2( S^1;M) <cit.>.
    
    In the simplest possible case where 
    g( S^1) is diffeomorphic to S^1, (<ref>) can be characterized elegantly in terms of the topological degree of g (we refer the reader to Appendix <ref> for a brief introduction to this notion). Indeed, according to Hopf's Theorem[The Hopf Theorem is a result within differential topology asserting that the topological degree stands as the sole homotopy invariant among continuous mappings taking values in a sphere.], in this case
    
    (<ref>)     (g) = 0.

    For data g∈ H^1/2( S^1;M) that are not continuous, such equivalence still holds, but one needs to interpret the degree in the sense of the VMO-degree of Brezis and Nirenberg <cit.>.



We now construct an explicit example of a radially symmetric constraint map with a discontinuity at the origin.


Given n=m≥ 3 and a radius a ∈ (0,1), we aim to find a map u_a∈ C^1,1(B_1∖{0}; B_a^c)∩ W^1,2(B_1;B_a^c) such that u_a = 𝕀 on ∂ B_1
and which solves
        
    Δ u_a = - |Du_a|^2 u_aχ_{|u_a|=a}  in B_1,
    
i.e., the Euler–Lagrange system for constraint maps with  obstacle B_a, see (<ref>).
   
   We look for a map u_aB_1∖{0}→ B_a^c of the form 
    
    u_a (x) := w_a(|x|) x/|x|.

The above system yields an overdetermined ODE for w_a, namely
     
    ẅ_a + n-1/rẇ_a = n-1/r^2 w_a    for r ∈ (r_a,1),
    
            w_a(r_a) = a,   ẇ_a(r_a) = 0,    w_a(1) = 1, 
    
            w_a(r)=a   for r ∈ [0,r_a],

    where r_a ∈ (0,1) is a free parameter corresponding to the free boundary constraint. 
Ignoring the  boundary conditions, the above ODE  has two  particular solutions, namely  r and r^1-n.
We thus  look for a solution of the form
  
    w_a (r) = 
        {[                    a     for r ∈ [0,r_a],; t_a r + (1-t_a)r^1-n    for  r ∈ (r_a,1), ].

    where the parameters t_a,r_a ∈ (0,1) satisfy

    t_ar_a+(1-t_a)r_a^1-n=a,     t_a+(1-n)(1-t_a)r_a^-n=0.

One can easily check that the system above has a unique solution. In addition, one can observe that r_a→ 0^+  as a→ 0^+, while r_a→ 1^- as a→ 1^-.  
Returning to the original map u_a,  the coincidence set is given by u_a^-1(∂ B_a) = B_r_a, with free boundary ∂ u_a^-1((B_a)^c) = ∂ B_r_a. 




An interesting problem is to understand whether the map u_a is a globally minimizing constraint map. We recall that the radial solution[Note that the function u_1(x)= x/|x| corresponds to the limit of the functions u_a as a→ 1^-.] u_1(x)= x/|x| of the Dirichlet problem for harmonic maps into a sphere with identity boundary values is a minimizer, i.e.

    [u_1]≤[v]     for all  v∈ W^1,2_𝕀(B^n;𝕊^n-1),

see e.g. <cit.> and the references therein; here, as before, we assume that n≥ 3.   However, despite the intuitive nature of this result, it is quite non-trivial to show the minimality of u_1.
  By analogy with the case of harmonic maps, it seems plausible  that the maps u_a above should be minimizers. 
  


We now give a few more different examples of minimizing constraint maps.


    As mentioned in the introduction, (energy-minimizing) geodesic curves in Riemannian manifolds-with-boundary are (minimizing) constraint maps, see e.g. <cit.> for a study of their properties.




    Consider a smooth boundary condition g∂Ω→∂ M,  suppose that the obstacle M^c is convex, and let u∈ W^1,2_g(Ω;M) be a minimizing constraint map. Then, according to <cit.>, see also Corollary <ref> below, the non-negative function ρ∘ uΩ→ [0,+∞) is subharmonic and vanishes on ∂Ω. Therefore, by the maximum principle, ρ∘ u=0 in Ω and thus u^-1(∂ M)=Ω. It follows that u is a minimizing harmonic map into ∂ M.


An alternative  geometric  proof (which avoids subharmonicity of ρ∘ u)  is to take any supporting plane H(y) = (y-y_0) ·ν_y_0, with y_0 ∈∂ M, with 
M ⊂{ H < 0 },  
and consider the restriction of the map to 
{ H ≥ 0}. Applying the minimum principle to 
the harmonic function H(u (x)) in the set 
{ H > 0 } allows us to conclude that u^-1(∂ M) = Ω.






 In analogy with parametric minimal surfaces, one can show that 
    if the convex hull of the image of the boundary satisfies (u(∂Ω )) ⊂ M then u(Ω )⊂ M.   
Indeed,  as in Example <ref>, the distance function to the convex hull is subharmonic (as u is harmonic in each component), so the maximum principle shows that the distance function vanishes everywhere, proving that u(Ω)⊂(u(∂Ω))⊂ M.


We conclude this section with a comparison between minimizing constraint maps and solutions to the scalar obstacle problem.



 §.§ Comparison with the scalar obstacle problem

Given a scalar function ∈ C^∞(Ω), the classical obstacle problem consists in minimizing the Dirichlet energy  among all functions v ∈ W^1,2(Ω) satisfying the constraint v≥φ in Ω. Minimizers of this problem are precisely the solutions of the equation

    Δ v = (Δφ)χ_{v=}  in Ω.

This problem and many variations of it have been the subject of intensive research over several decades and from various perspectives. Several influential works that have shaped the recent landscape of the scalar theory are, for instance,
<cit.>.



Here, we consider graphical obstacles and compare the solutions of the classical scalar obstacle problem to minimizing constraint maps. As we shall prove below, these two objects are always different unless the solution does not “actively” touch the obstacle.  Informally speaking, the reason for this difference comes from the fact that the graph of a solution to the obstacle problem is pushed upwards by the obstacle, while the hypersurface parametrized by a minimizing constraint map is pushed away from the obstacle.

To begin our analysis, 
we denote by M_ the subgraph of , i.e., M_ = {(y̅,y_n+1)∈^n+1: (y̅) < y_n+1}, fix a solution v∈ W^1,2(Ω) to (<ref>), and consider the vectorial map u∈ W^1,2(Ω;M_) defined by 

    u(x) := (x,v(x)).


Assume by contradiction that u is a minimizing constraint map. Since ∈ C^2(Ω), we have v∈ C_^1,1(Ω), whence u∈ C_^1,1(Ω;M_). Since Dv = D on Ω∩{v = } = Ω∩ u^-1(∂ M), we also have 

    D_α u = e_α + (D_α) e_n+1  on Ω∩ u^-1(∂ M).

for every α∈{1,2,⋯,n}, where e_α is the standard α-th basis vector in ^n+1. Thus, denoting by A_y(·,·) the second fundamental form of ∂ M_ (which is precisely the graph of ) at y, 

    A_u(Du,Du) = (-D,1)/1 + |D|^2Δ  on Ω∩ u^-1(∂ M),

with the specific choice u as in (<ref>)[In fact, setting ψ:Ω→^n+1 as the canonical parametrization of the graph, ∂ M_, of , (i.e., ψ(x) := (x,(x))) we obtain, for any 1≤α,β≤ n,

    A_ψ(D_αψ, D_βψ) = (-D, 1 )/1 + |D|^2D_αβ  in Ω.
] and using (<ref>).
Thus, if u is a minimizing constraint map, it follows from the Euler-Lagrange system (<ref>) as well as the definition (<ref>) of u that 

    (Δ v)e_n+1 = Δ u = -Δ/1 + |D|^2((D_α)e_α - e_n+1)χ_{v = }  in Ω;

here, as usual, we used the summation convention for the repeated index α running through {1,2,⋯,n}. Comparing (<ref>) with (<ref>), it follows that 

    |D|^2Δ = 0   a.e. on Ω∩{v=}.

Note that (<ref>) is a very strong restriction.
Indeed, since D^2φ=0 (and so Δφ=0) a.e. on the set {Dφ=0}, (<ref>) is equivalent to implies

    Δ = 0   a.e. on Ω∩{v=}.

Combining this information with (<ref>), we deduce that Δ v=0 a.e., thus v is harmonic.

In other words, except in the trivial case when  v is harmonic (which means that the obstacle is not acting on the function v), the graph of the solution to the obstacle problem does not coincide with the corresponding minimizing constraint map.









§ REGULARITY OF THE DISTANCE TO THE CONVEX HULL OF THE OBSTACLE


In this section, we establish the uniform regularity of the distance of a minimizing constraint map to the convex hull of the given obstacle. This result significantly improves <cit.>.
For convenience, we state the result below with Y being an arbitrary closed convex set containing  M^c, but the result is particularly relevant when Y is the convex hull of M^c.


Let M be a smooth domain in ^m with compact complement, Y  a closed convex set containing  M^c, and u∈ W^1,2(Ω;M) a minimizing constraint map with u _W^1,2(Ω)≤Λ. Then for every δ > 0 there exists a modulus of continuity ω, depending only on n, m, ∂ M, Λ, and δ, such that 

    _B_r(x_0)(u,Y) ≤ω(r)    ∀ r ∈(0,δ),

whenever B_2δ(x_0)⊂Ω. 


We begin by showing that the distance function is subharmonic.


 Let Y be a closed convex set containing  M^c.   Then the function (u,Y) is weakly subharmonic in Ω.



  Define the function w:Ω→,  w (x)  :=  (u(x)  ,Y). Since {w  > 0}∖Σ(u)⊂ u^-1(M)∖Σ(u), it follows by (<ref>) that  Δ u = 0 inside int({ w > 0})∖Σ(u). Hence, since y↦(y,Y) is convex (by the convexity of Y), we conclude that 
   Δ w ≥ 0   inside  int({ w > 0})∖Σ(u).
 As _Σ(u) ≤ n-3 by Theorem <ref> and w∈ W^1,2(Ω), by a capacity argument it follows that Δ w ≥ 0 weakly in Ω. 


As a corollary, we obtain an interior L^∞-estimate for u:


Under the hypothesis of Theorem <ref>
there is  a∈ M^c such that |u - a| ∈ L_^∞(Ω) and 

    u -  a _L^∞(Ω')≤ c M^c +  c u - a_L^2(Ω)/(Ω',∂Ω)^n/2

for every subdomain Ω'⋐Ω, where c depends only on n.  



    Since M^c is compact, there is a∈ M^c such that M^c⊂B_R(a), with R :=  M^c. Set Y := B_R(a), so that (u,Y) = (|u-a| - R)_+. By Lemma <ref>, (u,Y) is weakly subharmonic in Ω. The conclusion then follows from the local L^∞-estimate for weakly subharmonic functions. 


We also immediately obtain the following conclusion on the range of constraint maps:



Let u ∈ W^1,2(Ω;M) be a minimizing constraint map. If Y is the closed convex hull of u(∂Ω)∪ M^c, then u(x)∈ Y for a.e. x∈Ω.



This follows from the weak maximum principle by applying Lemma <ref> to (u,Y).



In what follows, we shall fix a closed convex set Y containing M^c. We shall denote by _0= _0(n,m,∂ M) the small energy threshold from Corollary <ref>. In addition, c will denote a generic positive constant depending at most on n, m, ∂ M, and Λ, for a given Λ > 1. 

The proof of Theorem <ref> will be based on the two subsequent lemmas. The first lemma asserts that if a map mostly takes values outside an η-neighborhood of the convex hull of the obstacle, then the mapping is uniformly regular in the interior of the domain. 




    Let u∈ W^1,2(B_4;M) be a minimizing constraint map such that u _W^1,2(B_4)≤Λ. For every η > 0, there exists a constant σ≡σ(n,m,∂ M,Λ,η)∈(0,1), such that if |{(u,Y) ≤η}∩ B_2|≤σ, then u ∈ C^1,1(B_1) with the estimate 
    
    u _C^1,1(B_1)≤ c,

    where c > 1 depends only on n, m, ∂ M, Y, and Λ. 







In view of the -regularity theorem, Theorem <ref>, it suffices to find, under the assumption of the statement, a radius r_0≡ r_0(n,m,∂ M,Λ) > 0, such that 

    E(u,x,2r_0) < _0^2    ∀  x_0∈ B_1.

It suffices to check this claim when Y is the convex hull of M^c, as the convex hull is the smallest (closed) convex set containing M^c. 

Suppose that the claim is not true for the convex hull Y, for some η > 0. Then, for each k=1,2,⋯, there exists a minimizing constraint map u_k ∈ W^1,2(B_4;M) such that u_k _W^1,2(B_4)≤Λ and 

    |{(u_k,Y) ≤η}∩ B_2|≤σ_k → 0,

but for some x_k∈ B_1 and r_k → 0^+ it holds

    E(u_k,x_k,2r_k) ≥_0^2.

By Lemma <ref>,  up to a subsequence u_k→ u strongly in W^1,2(B_3;^m) as k→∞, for some minimizing constraint map u∈ W^1,2(B_3;M). In particular, extracting a further subsequence if necessary, we have u_k→ u a.e. in B_3, so 

    (u_k,Y)→(u,Y)  a.e. in B_3.

Passing to the limit in (<ref>), we observe that

    (u,Y) ≥η  a.e. in B_2.

In particular, B_2∩ u^-1(∂ M) = ∅, which combined with  (<ref>) yields Δ u = 0 in B_2, and therefore

    Σ(u)∩ B_2 = ∅.

However, by (<ref>), it follows from Corollary <ref> that 
    E(u,x_0,0^+)≥_0^2 > 0,
 for some x_0∈B_1; by Corollary <ref> this implies x_0∈Σ(u), a contradiction to (<ref>). This finishes the proof. 



In the setting of Lemma <ref>, if σ is chosen sufficiently small (for instance, if σ≪η^n/2), then in fact Δ u =0 in B_1. Indeed, for such σ we must have (u,Y)>0 in B_1, since otherwise by (<ref>) we would have (u,Y)≤η in a ball of radius c√(η), contradicting our assumption.


Our second lemma shows the uniform decay of the distance map at a point of high energy.


    Let u∈ W^1,2(B_4;M) be a minimizing constraint map, with u _W^1,2(B_4)≤Λ. Then for every η >0, there exist radii 0 < s < r ≪ 1, both depending only on n, m, ∂ M, Λ, and η, such that   
    
    E(u,0,s) ≥_0^2     (u,Y) ≤η in B_r.




    Since u _L^2(B_4)≤Λ, we have (u,Y)_L^2(B_4)≤ c. By Lemma <ref>, (u,Y) is weakly subharmonic in B_4, therefore (u,Y) ≤ c_0c in B_3 for some c_0 = c_0(n), cf. also Corollary <ref>.


    Let η∈ (0,1/2c_0c) be given. Let ℓ be a positive integer, and suppose that 
    
    λ_ℓ := (u_ℓ,Y)_L^∞(B_4) > η,

    where u_ℓ(x):= u(2^-ℓx) is still  a minimizing constraint map in W^1,2(B_4;M). By the monotonicity of the normalized energy, Lemma <ref>, we have 
    
    E(u_ℓ,0,4) = E(u,0, 2^2-ℓ) ≤ E(u,0,4) ≤ 4^2-nΛ^2.
 Furthermore, since (u,Y)≤ c_0c in B_3, (u_ℓ,Y) ≤ c_0c in B_4 as 2^2-ℓ≤ 3, for any ℓ≥ 1.   
    
    Let c̅ be the C^1,1-bound in (<ref>), so c̅ depends only on  the parameters n, m, ∂ M and max{Λ^2,c_0c}, and choose σ to be the measure threshold in the same lemma, determined further by 1/2η. Since we assume E(u,0,s) ≥_0^2, we have E(u_ℓ,0,2^ℓ s) ≥_0^2. Hence, choosing s sufficiently small so that 
    
    2^2ℓ+1 s^2|B_1|c̅ < _0^2,

    there must exist a set F⊂ B_1 of positive measure such that |Du_ℓ| > 2c̅ on F. In particular, the C^1,1-estimate (<ref>) cannot hold for u_ℓ. Then it follows as the contraposition of Lemma <ref> that |{(u_ℓ,Y) ≤1/2η}∩ B_2| > σ. Combining this with (<ref>) gives us |{(u_ℓ,Y) ≤1/2λ_ℓ}∩ B_2| > σ.
    As (u_ℓ,Y) is weakly subharmonic in B_4 and (u_ℓ,Y) ≤λ_ℓ in B_4, the De Giorgi oscillation lemma (see e.g. <cit.>) now implies that 
    
    λ_ℓ+2 = (u_ℓ,Y) _L^∞(B_1)≤ (1-μ)^2λ_ℓ,

    for some μ∈ (0,1/2), depending only on n and σ_η, hence on n, m, ∂ M, Y, Λ and η. 

    Iterating the implication (<ref>)–(<ref>), and recalling that λ_ℓ≤λ_1 ≤ c_0c, we obtain 
    
    λ_ℓ≤max{ 4(1-μ)^ℓ c_0c, η},

    for every ℓ∈ for which (<ref>) holds. Taking ℓ̅:= min{ℓ∈: 4(1-μ)^ℓ c_0c ≤η}, and selecting s to be sufficiently small so that (<ref>) holds for all ℓ≤ℓ̅, we obtain λ_ℓ̅≤η, that is, 
    
    (u,Y)≤η  in B_4· 2^-ℓ̅.

    Finally, we choose r := 4· 2^-ℓ̅; it is not hard to see from (<ref>) that r > s. Tracking down the dependence of these radii, we also verify that both r and s depend only on n, m, ∂ M, Y, Λ and η as desired. 


We are now ready to prove the universal modulus of continuity of (u,Y).


It suffices to prove the theorem when Ω = B_4, x_0 = 0, δ = 1. The general case follows by a simple scaling argument. 

    Let η > 0 be given and let λ = 1/8. We fix a constant c(n)  to be fixed later, and we choose radii 0<s<r≪ 1 as in Lemma <ref>, corresponding to the parameters n, m, ∂ M, c(n)Λ, and η.     
    
    If E(u,0,2λ s) ≥_0^2, then Lemma <ref>, applied to u_2λ(x) := u(2λ x), shows that 0≤(u,Y) ≤η in B_2λ r, whence 
    
    _B_2λ r(u,Y)≤η.

    This application of Lemma <ref> determines the choice of c(n).

    Alternatively, if E(u,0,2λ s) < _0^2, then by the -regularity theorem, Theorem <ref>, we have u∈ C^1,1(B_λ s) and |Du|≤c_0/λ s in B_λ s, for some constant c_0≡ c_0(n,m,∂ M). Combined with the fact that (·,Y) is 1-Lipschitz, we find that
    
    _B_ηλ s/(2c_0)(u,Y)≤η.

    By virtue of (<ref>) and (<ref>), we derive in either case that the oscillation of (u,Y) is bounded by η in B_t(x_0), for some radius t ≡ t(n,m,∂ M,Λ,η). This finishes the proof. 


A direct consequence of Theorem <ref> is that, when the obstacle M^c is convex, there exists a universal continuity estimate on ρ∘ u, thus improving <cit.>.


    Let M⊂^m be a smooth domain with compact convex complement, and let u∈ W^1,2(Ω;M) be a minimizing constraint map with u _W^1,2(Ω)≤Λ. Then ρ∘ u is subharmonic and continuous in Ω. In addition, its modulus of continuity depends on n, m, ∂ M, Λ, and δ only.



 



§ UNIVERSAL L(-EPSILON)-ESTIMATE NEAR DISCONTINUITIES



In this section we establish a universal L^-γ-estimate of the energy density of minimizing constraint maps near effective discontinuity points,[By effective discontinuity points, we mean the points which have normalized energy above the -regularity threshold. The set of effective discontinuity points necessarily includes a neighborhood of the set Σ(u) of the actual discontinuity points, due to  the monotonicity of the energy. For further discussion, we refer the reader to <cit.>.] see Theorem <ref>.
As mentioned before, this result also applies to classical harmonic maps, and we believe it to be of its own interest.


Throughout this section, we shall fix _0 ≡_0 (n,m,∂ M) as the energy threshold provided by Corollary <ref>. Let us begin our analysis by finding the critical scale, labeled  r_u (x) below.




Let  u∈ W^1,2(B_4;M) be a minimizing constraint map satisfying u _W^1,2(B_4)≤Λ. 
There exists a large integer ℓ_0 > 1, depending only on n, m, ∂ M, and Λ, such that the following holds:  setting 

    r_u(x) := sup{ r∈ (0,2]: E(u,x,r)≤ℓ_0^-2_0^2}   if x∈ B_2∖Σ(u),
    
    0    if x∈ B_2∩Σ(u),

we have that Σ(u) = r_u^-1(0) and that if E(u,y,1)≥_0^2 for some y∈ B_2, then r_u(x) ≤ 1 for all x∈ B_2.


The assertion Σ(u) = r_u^-1(0) follows immediately from Corollary <ref>, as _0 is chosen as the energy threshold there.

Assume now that E(u,y,1)≥_0^2 for some y∈ B_2. To prove that r_u(x)≤ 1 for all x∈ B_2, we claim that 

    E(u,x,1) >1/ℓ_0^2_0^2,  ∀ x∈ B_2,

for some positive integer ℓ_0 ≡ℓ_0(n,m,∂ M,Λ). Once the claim holds, the conclusion follows from the monotonicity of the normalized energy, Lemma <ref>. 

Suppose that the claim does not hold. Then there is, for every k∈, a minimizing constraint map u_k ∈ W^1,2(B_4;M) with u_k _W^1,2(B_4)≤Λ such that E(u_k,y_k,1)≥_0^2 for some y_k ∈ B_2, but 

    E(u_k,x_k,1) ≤1/k^2_0^2,

for some x_k ∈ B_2. We may assume without loss of generality that x_k→ x_∞ and y_k→ y_∞ as k→∞, for some x_0,y_0∈B_2.
Up to a subsequence, the compactness of minimizing constraint maps (Lemma <ref>) implies that u_k converges strongly to u in W^1,2(B_3;M), where u is a minimizing constraint map. Given that B_1(y_k) ∪ B_1(y_∞) ⊂ B_3, this strong convergence implies that 

    E(u,y_∞,1) ≥_0^2 > 0.

However, since B_1(x_k)∪ B_1(x_∞)⊂ B_3, we also have

    E(u,x_0,1) = 0,

thus |Du| = 0 a.e. in B_1(x_0). 

Note now that, in view of (<ref>), we have |Δ u|≤ c|Du|^2. Also, u is locally Lipschitz in B_3∖Σ(u). Furthermore, Theorem <ref> implies that _Σ(u)≤ n-3 < n-1, and so by <cit.> the open set B_3∖Σ(u) is connected. Thus, the classical unique continuation principle (see <cit.>) yields |Du| = 0 everywhere outside Σ(u), whence also |Du|=0 a.e.  in B_3; in particular E(u,y_0,1) = 0,  contradicting (<ref>). This finishes the proof.




In what follows, we shall also fix the large constant ℓ_0 ≡ℓ_0(n,m,∂ M,Λ) chosen in Lemma <ref>. Given a smooth non-constant map fΩ→^m and a ball B_r(x_0)⊂Ω, let us write N(f,x_0,r) for the frequency of f in the ball B_r(x_0), i.e.,
 
    N(f,x_0,r) := r∫_B_r(x_0) |Df|^2 dx/∫_∂ B_r(x_0) |f - (f)_x_0,r|^2 ,

 where (f)_x_0,r denotes the integral average of f over the (n-1)-dimensional sphere ∂ B_r(x_0). When x_0 = 0, we shall write (f)_r ≡ (f)_0,r for simplicity. 

We are ready to make the key observation of this section: the frequency is uniformly bounded below the critical scale, provided that the mapping has discontinuities nearby (or, more generally, points of sufficiently large energy).
 
 

Let u∈ W^1,2(B_4;M) be a minimizing constraint map satisfying u _W^1,2(B_4)≤Λ. There exist a frequency threshold λ_0 > 1, depending only on n, m, ∂ M, and Λ, such that if E(u,y,1)≥_0^2 for some y∈ B_2, then 
     
    N(u,x,r_u(x)) ≤λ_0,

     for any x∈ B_1∖Σ(u), where r_u(x) is as in (<ref>).
 

  
Throughout the proof, c will denote a generic constant depending at most on n, m and ∂ M. Assume that the conclusion of the lemma is false. Then for every k ∈, there must exist a minimizing constraint map u_k ∈  W^1,2(B_4;M) such that u_k _W^1,2(B_4)≤Λ, E(u_k,y_k,1)≥_0^2 for some y_k∈ B_2, but at some x_k ∈ B_1∖Σ(u_k) it satisfies, with r_k := r_u_k(x_k),
     
    N(u_k,x_k,r_k) ≥ k.
    
By Lemma <ref>, r_k ≤ 1 for all k. This, along with x_k∈ B_2, yields 
     
    B_3/2r_k(x_k)⊂ B_3/2(x_k)⊂ B_4.

In what follows, let us write  
     
    v_k (y) := u_k(1/2r_ky + x_k),

 which is a minimizing constraint map in W^1,2(B_3;M). By Lemma <ref>, (<ref>) and the a priori bound ∫_B_1|Du_k|^2 dx ≤Λ^2, we deduce that
     
    E(v_k,0,3) = E(u_k,x_k,3/2r_k)≤ E(u_k,x_k,3/2) ≤ c∫_B_4 |Du_k|^2 dx ≤ cΛ^2.

In addition, since M^c is compact and u_k _L^2(B_4)≤Λ, Corollary <ref> implies that |u_k|≤ c (1+Λ) a.e. in B_7/2, thus
 
    |v_k|≤ c(1 + Λ)  a.e. in B_5/2.

In particular {v_k}_k=1^∞ is bounded in W^1,2(B_5/2;M), so Lemma <ref> yields a minimizing constraint map v ∈ W^1,2(B_2;M) such that 
     
    v_k→ v  strongly in W^1,2(B_2;^m),

after potentially extracting a subsequence. 

However, as (<ref>) yields N(v_k,0,2) = N(u_k,x_k,r_k) ≥ k, it follows from (<ref>) that
     
    ∫_∂ B_2 |v_k - (v_k)_1|^2 d^n-1≤c/kΛ^2,

for every k∈ .
By (<ref>), the boundedness of the trace operator <cit.>   implies 
     
    v_k → v   strongly in L^2(∂ B_2;^m),

and so we find, via (<ref>), that  
     
    v  = a  a.e. on ∂ B_2,

for some point a∈∂ M. 

We now observe that (<ref>) implies v = a a.e. in B_3: taking : B_3→^m defined by = a in B_2 and = v in B_3∖ B_2,  is an admissible map and so the minimality of v implies 

    ∫_B_2|Dv|^2 dx ≤∫_B_2 |D|^2 dx = 0,

i.e. |Dv| = 0 a.e. in B_2. As in the proof of Lemma <ref>, the  unique continuation principle <cit.> allows us to deduce that v = a in 
B_3∖Σ(v), which then readily implies that v = a in B_3 and Σ(v) = ∅. This is however a contradiction, as the choice of r_k and (<ref>) yield

    E(v,0,2) = lim_k→∞ E(v_k,0,2) = lim_k→∞ E(u_k,x_k,r_k) ≥1/ℓ_0^2_0^2.

This finishes the proof. 
 

Next we show that, above the critical scale, the differential of the map has a uniform lower bound on a universal fraction of each cube.

 
 
For any σ∈ (0,1), there are a small constant δ > 0, depending only on n, m, ∂ M, Λ, and σ, and a large constant c_0 > 1, depending solely on n, m, and ∂ M, such that for any minimizing constraint map u∈ W^1,2(B_8;M) satisfying u _W^1,2(B_8)≤Λ, if E(u,0,4) ≥1/ℓ_0^2_0^2, then
     
    |{|Du| ≥δ |Q|^-1/n}∩ Q| ≥ (1-σ)|Q|,

where Q=[-1/√(n),1/√(n)]^n is the largest cube contained in B_1.
 

 



Fix σ∈(0,1). Assume, for the sake of contradiction, that for each k∈, there exists a minimizing constraint map u_k ∈ W^1,2(B_8;M) satisfying u_k_W^1,2(B_8)≤Λ and E(u_k,0,4) ≥1/ℓ_0^2_0^2, but 
 
    |{ |Du_k| >δ_k |Q|^-1/n}∩ Q| < (1-σ)|Q|,

 along a sequence δ_k→ 0. 

 
 As in the proof of Lemma <ref>, one can find, via Lemma <ref>, a minimizing constraint map u ∈ W^1,2(B_4;M) such that 

    u_k → u   strongly in W^1,2(B_4;^m),

along a subsequence. To simplify our notation, let us assume without loss of generality that the convergence holds along the full sequence. 
Therefore, by Egorov's theorem, there exists a closed set F⊂ Q, with 

    |F| < 1/2σ|Q|,

such that 

    Du_k → Du   uniformly on Q∖ F.

Thus, by (<ref>) and (<ref>),

    |{ |Du_k| ≤δ_k |Q|^-1/n}∩ (Q∖ F)| ≥σ |Q| - |F| ≥1/2σ |Q|.

Recalling that δ_k→ 0, we obtain from (<ref>) and (<ref>) that 

    |{ |Du| = 0 }∩ (Q∖ F)}| ≥1/2σ |Q|.

Arguing as in the proof of Lemma <ref>, we again deduce from the minimality of u and the unique continuation principle that |Du| = 0 in B_4. However, by (<ref>),

    E(u,0,4) = lim_k→∞ E(u_k,0,4) ≥1/ℓ_0^2_0^2,

which is a contradiction. This completes the proof.

 


Next, we look below the critical scale r_u(x). There, our system (<ref>) can be considered as a linear system with  bounded coefficients for the first-order term, due to the -regularity theorem. Therefore, the energy density Du admits a uniform doubling property depending on the frequency of the mapping u.  For the reader's convenience, we recall these standard arguments in Lemma <ref> in the appendix.

 
 
Let λ > 1. Then there are constants δ = δ(n,m,∂ M,λ) and σ = σ(n,m,∂ M,λ), 0 < δ,σ < 1, such that if u∈ W^1,2(B_4;M) is a minimizing constraint map satisfying E(u,0,4) ≤_0^2 and N(u,0,4) ≤λ, then 
 
    |{ |Du| ≥δ Du_L^∞(3Q)}∩ Q|≥ (1-σ)|Q|,

 for every cube Q such that 3Q⊂ B_2. 
 

 
By our assumptions and Theorem <ref>, we deduce from (<ref>) that 
 
    Δ u^i = a_ij^α D_α u^j  in B_2,

with a_ij^α_L^∞(B_2)≤ c, where c depends only on n, m and ∂ M. Hence, the assertion follows immediately by applying Lemma <ref> with  f = u. 
 

 Combining the last three lemmas, we arrive at the following conclusion.

 
 
There are constants δ_0,σ_0 ∈ (0,1), both depending only on n, m, ∂ M, and Λ, such that for any minimizing constraint map u∈ W^1,2(B_4;M) satisfying u _W^1,2(B_4)≤Λ, if E(u,y,1)≥_0^2 for some y∈ B_2, then 
 
    |{ |Du| ≥δ_0 min{|Q|^-1/n,  Du_L^∞(3Q)}}∩ Q| ≥ (1-σ_0)|Q|,

 for every cube Q centered in B_2 such that Q≤1/8. 
 

 
We may assume without loss of generality that Λ > 1. Let λ_0 be as in Lemma <ref>, and choose δ_1 and σ_0 as in Lemma <ref> corresponding to λ =λ_0. We then select δ_2 as in Lemma <ref> corresponding to the chosen σ_0 (and with cΛ in place of Λ for some c≡ c(n,m,∂ M)). Finally, we set δ_0 := min{δ_1,δ_2} > 0. 

Let Q be a cube centered in B_2 with Q≤1/8. Denote by s its sidelength and by x_0 its center, so that Q = Q_s(x_0), √(n)s ≤1/8, and |x_0|≤ 2. Let B_r(x_0) be the smallest ball containing Q, i.e. r=1/2√(n)s ≤1/16. Then B_8r(x_0)⊂ B_1(x_0)⊂ B_3. Hence, the monotonicity of the normalized energy, Lemma <ref>, along with our assumption u _W^1,2(B_4)≤Λ, yields  
 
    E(u,x_0,8r)≤ E(u,x_0,2) ≤ 2^2-nDu_L^2(B_4)^2 ≤ 2^2-nΛ^2.

Moreover, by the local L^∞-estimate, Corollary <ref>, we have 

    u _L^∞(B_8r(x_0))≤ u_L^∞(B_3)≤ c + c u_L^2(B_4)≤ cΛ;

in the last inequality, we also used Λ > 1. 
Recall the critical scale r_u(x_0) defined in (<ref>). By Theorem <ref> and Lemma <ref>, it holds 0 ≤ r_u(x_0) ≤ 1.  
 Let us divide the proof into two cases.
 

 
 
 r ≥1/4r_u(x_0).
 
In view of (<ref>), we have E(u,x_0,4r) ≥1/ℓ_0^2_0^2. Also, by virtue of (<ref>) and (<ref>), we have 

    u_x_0,r_W^1,2(B_8)≤ cΛ,

where u_x_0,r(y) := u(ry + x_0) with c≡ c(n,m,∂ M). Hence, Lemma <ref> (with σ = σ_0 and δ = δ_2 there) applies to u_x_0,r. Rescaling back and recalling that Q is the largest cube contained in B_r(x_0), since δ_0 ≤δ_1 we obtain  
 
    |{ |Du| ≥δ_0 |Q|^-1/n}∩ Q| ≥ (1-σ_0)|Q|.

 Thus, the conclusion of the lemma follows.  


 
 r < 1/4r_u(x_0).
  
In this case, we have x_0∉Σ(u) and E(u,x_0,4r) < 1/ℓ_0^2_0^2. By Lemma <ref> we have N(u,x_0,r_u(x_0)) ≤λ_0, and by definition E(u,x_0,r_u(x_0)) ≤1/ℓ_0^2_0^2 < _0^2. Thus Lemma <ref> applies to u_x_0,r_u(x_0) (y) := u(r_u(x_0)y + x_0). Since 4r ≤ r_u(x_0), and Q⊂ B_r(x_0), we have 3Q⊂ B_3r(x_0)⊂ B_r_u(x_0)(x_0). Thus, we deduce (after rescaling back) that 
 
    |{ |Du| ≥δ_0Du_L^∞(3Q)}∩ Q| ≥ (1-σ_0)|Q|.

 Therefore, the conclusion of this lemma is verified for the alternative case as well. This finishes the proof. 
 

 We are now ready to prove Theorem <ref>, which is a direct consequence of the last lemma and the Calderón–Zygmund cube decomposition, whose statement we recall here for the reader's convenience (see e.g. <cit.> for a proof).

 
 Let A,B⊂ Q_1 be measurable sets, and let σ > 0 be given. Assume the following:  
 
 
  * |A| ≤σ.
 
  * If |A∩ Q| > σ |Q| for some dyadic cube Q, then Q̃⊂ B, where Q̃ is the predecessor of Q. 
 
 Then |A| ≤σ |B|. 
 



Consider a collection of disjoint open cubes Q^i, centered in B_2, with the same side-length s, i.e.  n s^2 = ( Q^i)^2, such that B_1⊂⋃_i=1^i_0Q^i and Q^i≤1/8 for each i. We can take the collection in such a way that its cardinality i_0 is bounded by a constant depending only on n. By the assumption Q^i≤1/8, we must have 64ns^2 ≤ 1. 

Take _0, ℓ_0, δ_0, and σ_0 as in Corollary <ref>, Lemma <ref> and Lemma <ref>. We claim that 
 
    |{|Du| ≤ t}∩ Q^i| ≤min{ct^2γ,1}s^n,

 for every t > 0, for some c > 1 and γ > 0, both depending only on n, m, ∂ M, and Λ, but not on i. Then by (<ref>) and elementary real analysis (integrating over all volume of super-level sets), we deduce that 
 
    ∫_B_1 |Du|^-γ dx    ≤∑_i=1^i_0∫_Q^i |Du|^-γ dx = γ∑_i=1^i_0∫_0^∞τ^-1 + γ |{ |Du| ≤τ^-1}∩ Q^i| dτ
       ≤γ i_0 s^n [∫_0^1 τ^-1+γ dτ + ∫_1^∞ cτ^-1-γ dτ] ≤ c,

where the last inequality follows from our choice (specified above) of both i_0 and s, in such a way that they are bounded above by a constant depending only on n.

To prove (<ref>),  let us define, for each k ∈ℕ, the set
 
    A_k^i := { |Du| ≤δ_0^k}∩ Q^i.

 It is clear that A_k+1^i⊂ A_k^i. Moreover, Lemma <ref>
 implies that |A_1^i|≤σ s^n, therefore 
 
    |A_k^i|≤σ s^n,   ∀ k ∈ℕ.

 Next we claim that if Q is a member of the dyadic cube decomposition of Q^i then 

    |A_k+1^i∩ Q| > σ_0 |Q|     Q̃⊂ A_k^i,
 where Q̃ is the predecessor of Q.
 
To prove this, assume by contradiction that Q̃∖ A_k^i≠∅. By the definition of A_k^i, we have 
 
    D u_L^∞(Q̃) > δ_0^k.

 Since Q̃⊂ 3Q ⊂ 3Q^i⊂ B_2 and |Q|^-1/n≥ |Q^i|^-1/n = s^-1 > √(n) > 1 > δ_0^k+1, Lemma <ref> yields that 
 
    |{ |Du| ≥δ_0^k+1}∩ Q| ≥ (1-σ_0)|Q|.

This implies that |A_k+1^i∩ Q| ≤σ_0 |Q|, a contradiction that proves the result.

Thanks to (<ref>) and (<ref>), it follows from Lemma <ref> that |A_k+1^i| ≤σ_0 |A_k^i|. Since k was an arbitrary positive integer and |A_1^i|≤σ_0s^n, we  conclude that 
 
    |A_k^i| ≤σ_0^k s^n=s^n (δ_0^k)^2γ,  ∀  k∈,

 with γ:=logσ_0/2logδ_0.
Due to this bound, (<ref>) follows. Since both δ_0 and σ_0 depend only on n, m, Λ, and ∂ M, so do the constants γ and c. This completes the proof. 
 

 







 


§ REGULARITY NEAR FREE BOUNDARIES


The purpose of this section is to prove Theorem <ref>, which asserts that,  whenever the obstacle is uniformly convex, minimizing constraint maps are C^1,1 in a universal neighborhood of the non-coincidence set. Therefore, throughout this section, we assume that M is a smooth domain with uniformly convex complement. Moreover, we fix _0 as the energy threshold chosen as in Lemma <ref>, which depends only on n, m, and ∂ M. 



As we shall see, the result follows from Theorem <ref>, proved in the previous section,  with the following elementary   De Giorgi-type lemma, which we will apply to the subharmonic function ρ∘ u.

 
 Let w∈ W^1,2(B_2) be a nonnegative subharmonic function, and suppose that there are positive constants a, p, q such that 
 
    |{ w > 0}∩ B_r|≤a/(R-r)^q w _L^∞(B_R)^p,

 whenever 1≤ r <R ≤ 2. Then there is a constant η∈(0,1), depending only on n, a, p, and q, such that if w ≤η a.e. in B_2, then w = 0 a.e. in B_1. 
  

 
 Let r_k := 1 + 4^-k and s_k := 1 + 2· 4^-k-1, for each k∈. Then 
 r_k - s_k = 2· 4^-k-1 and s_k - r_k+1 = 4^-k-1. Denote δ_k := |{ w > 0}∩ B_s_k| and λ_k := w_L^∞(B_r_k). Then our hypothesis (<ref>) yields
 
    δ_k ≤ a (2· 4^k+1)^q λ_k^p.

This, combined with the local L^∞-estimate for subharmonic functions, implies that 
 
    λ_k+1≤c/(s_k-r_k+1)^nw_L^2(B_s_k)≤ c· 4^n(k+1)λ_k δ_k ≤  C a · 4^ k(n+q)λ_k^1+p,

 where c= c(n) and C=2^3 β + 2n c. Thanks to this recursive relation, it follows that
 
    λ_0 ≤η  ⟹  lim_k→∞λ_k = 0,

 if η is sufficiently small; the smallness condition can be determined a priori by n, a and γ only. Since w _L^∞(B_1)≤λ_k for all k, we conclude that w = 0 a.e. in B_1.
 


We can now prove the following key step towards Theorem <ref>. 


There exists a constant η>0, depending only on n, m, ∂ M, and Λ, such that for any minimizing constraint map u ∈ W^1,2(B_4;M) satisfying ∫_B_4 |Du|^2 dx ≤Λ^2 and ρ∘ u ≤η in B_4, 

    E(u,y,1)≥_0^2 for some y∈ B_2     B_1⊂ u^-1(∂ M).

 

 
Fix a pair of radii with 1 ≤ r<R ≤  2, and let ψ∈ C_0^∞(B_R) be a smooth cutoff function such that ψ = 1 in B_r, |D^2ψ|≤ c/(R-r)^2, and 0≤ψ≤ 1 in B_R, with c = c(n). In view of the equation (<ref>) and by the geometric bounds (<ref>) with ξ = D_α u(x) (note that, by chain rule, ∇Π(u(x)) · D_α u(x) = D_α (Π∘ u)(x)), we find
 
    c/(R-r)^2∫_B_R (ρ∘ u) dx    ≥∫_B_R (ρ∘ u)Δψ dx 
       = ∫_B_R∩ u^-1(M)ψρ_u(Du,Du) dx 
       ≥ c_0∫_B_r∩ u^-1(M) |D(Π∘ u)|^2  dx.

On the other hand, as ρ∘ u is weakly subharmonic in B_4 due to the convexity of M^c, the Caccioppoli inequality along with ρ∘ u ≤η < 1 in B_4 and |D(ρ∘ u)| = 0 a.e. in B_r∖ u^-1(M) shows that 

    c/(R-r)^2∫_B_R (ρ∘ u) dx ≥∫_B_r∩ u^-1(M) |D(ρ∘ u)|^2 dx.

Adding (<ref>) and (<ref>), and utilizing |Du|^2≤ c|D(Π∘ u)|^2 + c|D(ρ∘ u)|^2 a.e., we get

    c/(R-r)^2∫_B_R (ρ∘ u) dx ≥ c_0 ∫_B_r∩ u^-1(M) |Du|^2 dx,

with a possibly larger constant c>1. 

Let ,η >0 and c_1 > 1 be the constants determined a priori by n, m, ∂ M, and Λ, as in Theorem <ref>. Since by assumption E(u,y,1)≥_0^2 for some y∈ B_2 and ρ∘ u≤η in B_4, we invoke Theorem <ref> to obtain 
 
    ∫_B_2 |Du|^-γ dx ≤ c.

 Choosing p > 1 such that 1/p = 1/2 + 1/γ, Hölder inequality and (<ref>) yield
 
    |B_r∩ u^-1(M)|^2/p≤(∫_B_r∩ u^-1(M) |Du|^-γ dx)^2/γ∫_B_r∩ u^-1(M) |Du|^2 dx ≤ c^2/γ∫_B_r∩ u^-1(M) |Du|^2 dx.

 By (<ref>) and (<ref>), this implies
 
    |B_r∩ u^-1(M)| ≤c/(R-r)^pρ∘ u_L^∞(B_R)^p/2;

 note that p = 2γ/2+γ depends only on n, m, ∂ M, and Λ. Since u^-1(M) = {ρ∘ u > 0}, condition  (<ref>) is verified. By choosing η even smaller if necessary,  Lemma <ref> applies, so from the assumption ρ∘ u ≤η in B_4 we conclude that ρ∘ u = 0 in B_1. In other words,  B_1⊂ u^-1(∂ M), as desired. 
 


 With the help of Lemma <ref>  we establish the following rigidity theorem, which may be of independent interest.

 
 Let u∈ W_^1,2∩ L^∞(^n;M) be a locally minimizing constraint map. Suppose that 
    lim sup_k→∞[ k^2-n∫_B_k  |Du|^2  dx ] < + ∞  and   0∈Σ(u) .

 Then u ∈ W_^1,2(^n;∂ M) and in particular u is a locally minimizing harmonic map into ∂ M. 
 


  
By Lemma <ref>, Corollary <ref>, 0∈Σ(u), and (<ref>), there is a constant θ_0 > _0^2 such that 
 
    lim_k→∞[ (kR)^2-n∫_B_kR |Du|^2 dy ] = θ_0,

 for every R> 0.  Let us write u_k (y) := u(ky), which is a minimizing constraint map in W^1,2(B_8;M). By (<ref>) and the bound |u|∈ L^∞(^n), the sequence {u_k}_k=1^∞ is bounded in W^1,2∩ L^∞(B_8;M). Thus, by Lemma <ref>, there exists a minimizing constraint map u_∞∈ W^1,2(B_4;M) and a subsequence k_i→∞ such that u_k_i→ u_∞ strongly in W^1,2(B_4;^m).  Moreover, by Lemma <ref>, u_∞ is 0-homogeneous in B_4 since, by (<ref>), we have 
 
    R^2-n∫_B_R |Du_∞|^2 dy = lim_i→∞[ R^2-n∫_B_R |Du_k_i|^2 dy ] = lim_i→∞[ (k_i R)^2-n∫_B_k_i R |Du|^2 dx] = θ_0,

 for a.e. R∈(0,4). Since 0∈Σ(u_k), it follows from Corollaries <ref> and <ref> that 0∈Σ(u_∞). 

 As u_∞ is a minimizing constraint map, it follows from Corollary <ref> that ρ∘ u_∞ is weakly subharmonic and continuous in B_4. Then the 0-homogeneity of ρ∘ u_∞ implies that ρ∘ u_∞ is constant in B_4, cf. <cit.>. Since 0∈Σ(u_∞), in fact we must have
 
    ρ∘ u_∞ = 0  in B_4.

Indeed, if ρ∘ u_∞ =a for some a>0,  then we would have |u_∞^-1(∂ M)| = 0,  and so by (<ref>), Δ u_∞ = 0 in B_4, a contradiction to 0∈Σ(u_∞). We remark that this argument can also be found in the proof of <cit.>.

 Owing to (<ref>) and the strong convergence of u_k_i→ u_∞ in L^2(B_4;^m), we have ρ∘ u_k_i→ 0 in L^2(B_4). By the subharmonicity of ρ∘ u_k_i from Corollary <ref>, we have 
 
    ρ∘ u_k_i→ 0  in L^∞(B_4).

 Now we choose η to be a small constant, depending only on n, m, ∂ M, and Λ, as in Lemma <ref>. Then by (<ref>) we can find a large i_0 such that ρ∘ u_k_i≤η in B_4 for all i≥ i_0. Recall  that 0∈Σ(u_k_i), which implies E(u,0,1)≥_0^2 by Lemma <ref> and Corollary <ref>. Hence  Lemma <ref> implies that B_1 ⊂ u_k_i^-1(∂ M). Rescaling back, we have proved that 
 
    B_k_i⊂ u^-1(∂ M),

 for every i≥ i_0. Since k_i→∞, we deduce that u^-1(∂ M) = ^n, which proves u∈ W_^1,2(^n;∂ M). Finally, u is a minimizing harmonic map, since  W^1,2(D;∂ M)⊂ W^1,2(D;M) for any bounded domain D⊂^n. 
 


We are now ready to prove the main result of this section.


As M^c is convex, we can take Y = M^c in Theorem <ref>, which along with u _W^1,2(B_2)≤Λ yields a modulus of continuity ω, determined a priori by n, m, ∂ M, and Λ, such that for every r∈(0,1/2) and x_0∈ B_1, 

    _B_r(x_0) (ρ∘ u)≤ω(r).

By Corollary <ref>, the free boundary B_1∩∂ u^-1(M) is well-defined, so let us take an arbitrary point x_0 there. Now let η be as in Lemma <ref> (with cΛ^2 in place of Λ there), and take δ= δ(n,m,∂ M,Λ) ∈ (0,1/8) small such that ω(8δ) ≤η. By (<ref>) and the fact that ρ∘ u(x_0) = 0, we have 

    ρ∘ u ≤η  in B_8δ(x_0).

On the other hand, since B_8δ(x_0)⊂ B_1(x_0)⊂ B_2, Lemma <ref> yields

    E(u,x_0,8δ)≤ E(u,x_0,1) ≤∫_B_2 |Du|^2 dx ≤Λ^2.

Now, as x_0∈ B_1∩∂ u^-1(M), B_2δ(x_0)∖ u^-1(∂ M) ≠∅. By (<ref>) and (<ref>), the contraposition of Lemma <ref> (applied to u_x_0,2δ(y) := u(2δ y + x_0)) implies that 

    E(u,x_0,2δ) < _0^2.

Then, by Theorem <ref>, u∈ C^1,1(B_δ(x_0)), and since δ≡δ(n,m,Λ,∂ M), we have the estimate

    D^j u_L^∞(B_δ(x_0))≤ c,   j∈{1,2},

where c ≡ c(n,m,Λ,∂ M). 
This proves the validity of a uniform C^1,1 bound in a neighborhood of the free boundary.
On the other hand, since Δ u = 0 in B_1∩ u^-1(M), for any y_0∈ B_1∩ u^-1(M) with (y_0,B_1∩∂ u^-1(M)) > δ, we may apply the interior estimates for harmonic functions, and use the universality of δ as well as ∫_B_2 |Du|^2 dx ≤Λ^2, to obtain that 

    |D^j u(y_0)| ≤ c,    j∈{1,2}.

Combining the above two inequalities, we arrive at the desired conclusion. 








§ DISCONTINUITIES ON FREE BOUNDARIES PRODUCED BY DEGENERATE OBSTACLES



This section stems from the search for the minimal geometric conditions on the obstacle under which there are no discontinuities of u near the free boundary. In Theorem <ref> we proved the absence of discontinuities under the assumption that the obstacle is uniformly convex.
The main result in this section is Theorem <ref>, which shows the sharpness of the previous result: for convex obstacles, there may be discontinuities on the free boundary if we allow the obstacle to have flat sides. More precisely, we shall hereby call a subset T⊂∂ M a flat piece if T is relatively open, connected, and all the principal curvatures of T vanish.  Equivalently, T is a relatively open connected subset of a hyperplane.  


Before beginning the main proofs of this section, we collect some useful remarks.



Consider the setting of Theorem <ref>. By the boundary regularity theory <cit.>, whenever both ∂Ω and u|_∂Ω are of class C^2, we have that u ∈ C^1,α({x∈Ω: (x,∂Ω)< δ}) for  every α∈(0,1), where δ >0 depends only on n, α, the C^2-character of ∂Ω, and u|_∂Ω; in particular, 

    (Σ(u),∂Ω)≥δ.

Now, if M^c is convex then Ω∩ u^-1(M)  is an open set according to Corollary <ref>, and hence Δ u =0 in Ω∩ u^-1(M). In particular, we have u∈ C^∞(Ω∩ u^-1(M)). In conclusion, we obtain 

    u∈ C_^1,α(Ω∩ u^-1(M))∩ C^∞(Ω∩ u^-1(M)).



The previous remark shows that there are no discontinuities in a neighborhood of the fixed boundary ∂Ω. As we now observe,  discontinuities are unavoidable in the interior.


As mentioned in the introduction, constraint maps, just as harmonic maps, generally develop discontinuities for topological reasons. For the convenience of the reader, we briefly recall why this is the case. Given Ω⊂^n, we  have the following topological fact:

    u∈ C^0(Ω)
    
        u=𝕀 on ∂Ω    Ω⊆ u(Ω).

In particular, if M^c⋐Ω and u∈ W^1,2(Ω;M) is a minimizing constraint map with identity boundary values, then the conclusion in (<ref>) does not hold, and so we must have Σ(u)≠∅. This is essentially the setting of Theorem <ref>.

Implication (<ref>) is most easily proved using the notion of topological degree, which is  a way of counting (with multiplicity) the number of solutions x∈Ω to an equation

    u(x)=y,

The topological degree shows that in the setting of Theorem <ref>, we must have Σ(u)≠∅, but it can also be used to yield information about the image of u, since  y∈ u(Ω) whenever (u,Ω,y)≠ 0. 




In the setting of Theorem <ref>, we have


    u|_Ω∩ u^-1(M) is surjective onto Ω∩ M.

The proof of this fact is somewhat involved and relies on degree theory. In fact, in Proposition <ref> below we will prove a much stronger degree theoretic result, which we believe to be independently significant. We also note that, from well-known properties of the topological degree, see in particular Theorem <ref>(<ref>), our assumption (g) ≠ 0 on the boundary data implies that g is surjective, namely g(∂Ω)=∂Ω.


Assuming for now that (<ref>) holds,  let us show how it implies Theorem <ref>.  First, we state a simple consequence of the maximum principle:


Let u∈ W^1,2(Ω;M) be a minimizing constraint map, and let M and Ω be as in Theorem <ref>.  If u=g on ∂Ω and g(∂Ω)⊆∂Ω,  then the set Ω∩ u^-1(M) is open and connected.




Note that our assumptions  ensure that (∂Ω,∂ M) > 0.  Thus, since u = g on ∂Ω and g(∂Ω) ⊆∂Ω,  we see that the open set u^-1(M) 
contains a neighborhood of ∂Ω, according to the continuity provided by Remark <ref>. 

If the set u^-1(M) was not connected, then there would be a connected component C⊂ u^-1(M) that is disjoint from ∂Ω, thus ∂ C⊂∂ u^-1(M)∩Ω. We would then have (u,∂ M)=0 on ∂ C and (u,∂ M)>0 on C. However, since the function (u,∂ M) is subharmonic in u^-1(M) (since u is harmonic in u^-1(M) and (·,∂ M) is convex), this would contradict the weak maximum principle.





Let T be a flat piece of ∂ M, that is, 

    T⊂{x∈^n:e· x=c}
  for some e∈^n and c∈,
and assume without loss of generality that {e· x> c}⊂ M. Let M'⊂ M be the open half-cylinder with base T, namely

    M':={x=t e+x̅∈^n: x̅∈ T  and  t> 0 }.

Fix any point y ∈ T.  According to the surjectivity asserted in (<ref>) (see Proposition <ref> below),  there is a sequence {x_k}_k=1^∞⊂ u^-1(M)∩Ω such that u(x_k)→ y as k→∞. We may also assume that x_k→ x_∞ for some x_∞∈u^-1(M)∩Ω. We claim that

    x_∞∈Σ(u);

once this is shown, we must have x_∞∈∂ u^-1(M), since u is harmonic in the open set u^-1(M) according to (<ref>) and Theorem <ref>. 

Suppose, for the sake of contradiction, that x_∞∉Σ(u),  so there is r>0 with B_r(x_∞)⊂Ω∖Σ(u). Since T is a relatively open set in ∂ M and u∈ C^0(B_r(x_∞)), we have u(B_r(x_∞))⊂ M'∪ T, by taking r smaller if necessary. In particular, (<ref>) shows that u is harmonic in B_r(x_∞). Hence, 
e· u ≥ c  and Δ(e· u) = 0  in  B_r(x_∞).
Also, we have

    e·  u(x_∞) = lim_k→∞  e· u(x_k) = e· u(y) = c,
 so the strong minimum principle asserts that e · u = c in B_r(x_∞).  


Since e· u is a harmonic function in Ω∩ u^-1(M) and Ω∩ u^-1(M) is connected (see Lemma <ref>),  the unique continuation principle shows that e· u=c in Ω∩ u^-1(M).  On the other hand, u is continuous near ∂Ω and u(∂Ω)=g(∂Ω)=∂Ω (since (g)≠ 0, see Remark <ref>), so a contradiction is reached.  
 




The rest of this section is dedicated to proving  (<ref>). In fact, we will show a much more general result concerning the topological degree of minimizing constraint mappings, valid for non-convex obstacles, such as the one in Figure <ref>. We believe this more general result is independently interesting, and it is relevant in Corollary <ref> below.





Let Ω⊂^n be a smooth uniformly convex domain, and  M⊂^n be such that M^c is connected and M^c⋐Ω. Let O be the complement of the convex hull of M^c. Consider g∈ C^2(∂Ω;∂Ω) with (g) ≠ 0 and let u ∈ W^1,2_g(Ω;M) be a minimizing constraint map. 

For any domain O'⋐ O,  we have

    (u,Ω',·) = (g)   on Ω∩ O',     where Ω' := Ω∩ u^-1( O').

In particular,  u|_Ω∩ u^-1(O) is a surjective componentwise-harmonic map onto Ω∩ O.



Before beginning the proof of Proposition <ref>, we prove a simple topological lemma. This lemma does not rely on the specific geometric structure assumed in Proposition <ref>, and we use only that O⊆ M.



Let uΩ→M be a map such that u|_u^-1(O)  is continuous and u(∂Ω)⊆∂Ω. For every O'⋐ O we have u(∂Ω')⊆∂Ω∪∂  O', where Ω'=Ω∩ u^-1(O') is as in (<ref>).





Note that ∂Ω'= ∂Ω∪ [Ω∩∂ u^-1(O')]. Since we assume u(∂Ω)⊆∂Ω, it suffices to prove that 
    u(Ω∩∂ u^-1(O'))⊆∂  O'.
 For this purpose,  given x_0 ∈Ω∩∂ u^-1(O'), there is a sequence x_k∈Ω∩ u^-1(O') such that x_k→ x_0, and since u(x_k)∈ O' we have u(x_0)∈O'.

Assume by contradiction that u(x_0) ∈  O'. Then x_0 ∈Ω∩ u^-1(O'), and since Ω∩ u^-1(O') is an open set, there is a ball B_r(x_0)⊂Ω∩ u^-1(O'), impossible.

This proves that u(x_0)∈∂ O', as desired. 





Throughout the proof, we write f for the signed distance function to ∂Ω, with f<0 in Ω.  Since Ω is smooth and uniformly convex,  there is a tubular neighborhood (∂Ω) of ∂Ω where f is smooth and satisfies 

    f(ξ,ξ)≥ c|ξ^⊤|^2   on (∂Ω),
 for some c>0 and all ξ∈^n. Here we write ξ^⊤:= ξ - (ν·ξ)ν as in (<ref>), where now ν (with a slight abuse of notation deviating from the introduction) is the outwards unit normal to ∂Ω. For δ>0 small enough, we write

    O_δ := {-δ < f≤ 0}⊆Ω∩(∂Ω).

We split the proof into four steps.




There is no ball B⊂ u^-1(O) such that f∘ u=0 in B.


As in the proof of Theorem <ref>, since we assume that (g)≠ 0 we have that g is a surjective map, i.e. g(∂Ω)=∂Ω.
Now assume towards a contradiction that f∘ u=0 in some ball B⊂Ω∩ u^-1(O). Since  Δ u=0 in Ω∩ u^-1(O), we have

    0 = Δ(f∘ u) =    f_u (Du,Du)    in  B.

Since u(B)⊆∂Ω, Du lies in the tangent bundle T(∂Ω), i.e. Du = (Du)^⊤. Thus, by (<ref>)  we deduce that Du=0 in B. According to Lemma <ref>, the open set Ω∩ u^-1(O) is connected, and as u is harmonic there, 

by unique continuation
we must have that u is constant in Ω∩ u^-1(O). This contradicts the fact that u(∂Ω)= g(∂Ω)= ∂Ω.




    For every sufficiently small δ>0, there exists η > 0 such that Ω∩ u^-1(O_η) ⊂Ω∩ O_δ. 


Suppose, for the sake of contradiction, that there are some small δ>0 and a sequence η_k→ 0 for which points y_k∈∂ O_η_k and x_k∈ u^-1(y_k)∩Ω∖ O_δ can be found. Without loss of generality, we can assume that x_k→ x_∞ and u(x_k)=y_k→ y_∞.  Clearly, x_∞∈Ω∖ O_δ and y_∞∈∂Ω = f^-1(0). Since Theorem <ref> applies with Y=O^c, we have 

    (u(x_∞),∂ O)    =(u(x_∞),O^c) =lim_k→∞(u(x_k),O^c)
       =lim_k→∞(u(x_k),∂ O) 
    = (y_∞,∂ O)≥(∂Ω,∂ O)>0,

and so x_∞∈Ω∩ u^-1(O). In particular,  u is continuous at x_∞ and u(x_∞)=y_∞. Since f is convex in (∂Ω),  by continuity and harmonicity of u in Ω∩ u^-1(O) we conclude that the function f∘ u is subharmonic in a ball B centered at x_∞.  However,  Corollary <ref> shows that

    f∘ u ≤ 0   in Ω

therefore x_∞ is an interior maximum for f∘ u in B, since f∘ u(x_∞)=f(y_∞)=0.  Therefore, by the strong maximum principle we deduce that f∘ u = 0 in B.  This contradicts Step 1,  completing the proof of Step 2.


There is d∈ such that    (u,Ω',·) = d on Ω∩ O' for every O'⋐ O.


Since u∈ C^0(Ω∩ u^-1(O)) by Remark <ref>,  we have u ∈ C^0(Ω') (recall Ω' = Ω∩ u^-1(O')).  By Lemma <ref> we have Ω∩ O' ⊂^n∖ u(∂Ω') and so,  by Lemma <ref>, Ω∩ O' is contained in a connected component of ^n∖ u(∂Ω').  By Theorem <ref>(<ref>), (u,Ω',·) is constant in each connected component of ^n∖ u(∂Ω'), and so there is d∈ such that

    (u,Ω',·) = d    in Ω∩ O',

as desired. 


We have d=(g). 


Note that f∘ u  is subharmonic in a neighborhood of ∂Ω inside Ω, and it achieves a maximum on ∂Ω.  Thus, the Hopf lemma combined with Step 1 shows that 

    D_ν (f∘ u) ≥ 2 c   on ∂Ω,
 for some constant c>0, where we recall that ν is the outwards unit normal to ∂Ω, extended to (∂Ω) via the formula ν=∇ f.  Since f and u are C^1 near ∂Ω,  we deduce that 

    D_ν(f∘ u) ≥ c   on Ω∩ O_δ,

provided that δ>0 is sufficiently small.  

Consider an oriented orthonormal basis ℬ_x:={τ^1_x,…,τ^n-1_x,ν_x} at a point x∈∂Ω, and write 𝒯_x :=I-ν_x⊗ν_x for the matrix with columns τ_x^1,…, τ_x^n-1. 
Recall that (g) is an integer which can be computed as follows: if y∈∂Ω is a regular value[Recall that y is a regular value of g if the differential dg_x is non-singular at all points x∈ g^-1(y). In particular, by the inverse function theorem, a regular value of g has only a finite number of pre-images.] of g and g^-1(y)={x_1,…, x_i_0} for some points {x_i}⊂∂Ω, then

    (g) = ∑_x_i∈ g^-1(y)( dg_x_i),

where we see dg_x_i T_x_i∂Ω→ T_y∂Ω as an invertible linear map between (n-1)-dimensional vector spaces. More precisely, 

since u=g on ∂Ω, for each x ∈∂Ω we have

    dg_x = 𝒯_u(x)^t Du(x)𝒯_x,

where the superscript “t” denotes the transpose.

Moreover, there are  relatively open subset U_i, V⊂Ω such that x_i∈ U_i and y∈ V, 

    g^-1(V∩∂Ω)=∪_i=1^i_0 U_i∩∂Ω ,   where  g U_i ∩∂Ω→ V∩∂Ω is a diffeomorphism,

for all i. Note that, on ∂Ω, we also have

    ν_u(x)^t D u(x) 𝒯_x = D_τ(f∘ u)(x)=0,     ν_u(x)^t Du(x) ν_x = D_ν(f∘ u)(x).

Thus, for x∈∂Ω, we can write Du(x) with respect to the bases ℬ_x and ℬ_u(x) as follows:

    Du(x) = 
    [         dg_x            *;            0 D_ν(f∘ u)(x) ],

where * denotes an unspecified entry of the matrix.
Since u is C^1,α near ∂Ω, by shrinking δ, U_i, and V if needed,  we see from (<ref>) that

    ( Du(x))=( dg_x_i)   for all  x∈ U_i∩Ω.
  

Let η be determined in terms of δ through Step 2. By shrinking U_i and V further if necessary, we can also assume that V⊂Ω∩ O_η, so u^-1(V)⊂Ω∩ O_δ, and  by (<ref>) and (<ref>) we have

    u^-1(V) = ∪_i=1^i_0 U_i,      u U_i → V  is a diffeomorphism.

Thus any point y'∈ V∩Ω is a regular value for u which is not in u(∂Ω)=∂Ω and so

    (u,Ω',y')=
    ∑_x̃_i∈ u^-1(y')( Du(x̃_i))=
     ∑_x_i∈ g^-1(y)( dg_x_i)= (g)

for every >0 sufficiently small, as wished.




For the surjectivity claim, note that Theorem <ref>(<ref>) and the previous two steps show that Ω∩ O'⊆ u(Ω') for all O'⋐ O. The conclusion follows by taking the union over O'.



We conclude this section by noting that, for general non-convex obstacles, the non-coincidence set u^-1(M) is not necessarily open:


Let Ω,M,O and u be as in Proposition <ref>. Assume that M^c is non-convex and that ∂ O contains a flat piece T. Suppose in addition that

    u(Ω)⊆O.

Then u^-1(M) is not an open set.



Note that, by (<ref>), u∈ W^1,2_g(Ω;O) is a minimizing constraint map. Since O^c is convex, according to Theorem <ref> there is a point x̅∈∂ u^-1(O)∩Σ(u) and a sequence of points x_k→x̅ such that u(x_k)∈ O⊂ M  and u(x_k)→y̅ for some y̅∈∂ O∖∂ M. In particular, since u is continuous over u^-1(O), we have 

    lim inf_r→ 0(u,∂ M)_L^∞(B_r(x̅)) >0,

thus x̅∈ u^-1(M). On the other hand,  since x̅∈Σ(u), there is no ball around x̅ that is contained in u^-1(M), as otherwise u would be harmonic near x̅, according to (<ref>). Thus, u^-1(M) is not open.



We note that assumption (<ref>) is satisfied for many non-convex obstacles, such as the one in Figure <ref>. Indeed, for such an obstacle the nearest-point projection map π M→ O is 1-Lipschitz and so, by comparing a minimizing constraint map u∈ W^1,2_g(Ω;M) with a competitor π∘ u, we can deduce that in fact u∈ W^1,2_g(Ω;O).







§ BRANCH POINTS AND FREE BOUNDARIES





 §.§ General remarks on branch points

Our goal now is to understand the interaction between branch points and the free boundary. We begin by collecting some general remarks. Similarly to (<ref>), we state the following definition.



Let uΩ→^m be a minimizing constraint map.  We call x_0∈Ω∖Σ(u) a branch point of u, and we write x_0∈(u), if Du(x_0) does not have full rank. This leads to a decomposition

    (u) = ⋃_k=0^min{m,n}-1_k(u),    _k(u) := {x∈Ω∖Σ(u): rank(Du(x))=k }.

Given x_0∈(u), we say that:


  * x_0 is a false branch point if u(B_r(x_0)) is a C^1 manifold, for all r>0 sufficiently small;

  * x_0 is a true branch point if it is not a false branch point.



In order to illustrate the above concepts, consider the maps 

    u_k≃^2→^3,    z↦  (z^2,Re z^k),

which are component-wise harmonic. In particular, if M⊂^3 is a smooth domain such that u()⊆M, then u is a minimizing constraint map in W^1,2(Ω;M) for any Ω⊂. 
These maps also satisfy (u_k)=_0(u_k)={0},  and moreover 0 is a true branch point if k is odd, and it is a false branch point if k is even, see Figure <ref> for an illustration.



Before proceeding further, we compare Definition <ref> with other definitions of branch points occurring in the literature; we believe that this will also clarify the nature of our definition.

 In the function theory literature <cit.>, one typically considers maps between domains of the same dimension, i.e., m = n. The branch set ^(u) is defined as the set of points near which a map is not a local homeomorphism.  Clearly, for a C^1 map u we have

    ^(u)⊆(u) ,

according to the Inverse Function Theorem. This inclusion is generally strict, and this highlights the fact that our definition of branching is analytical, rather than topological. In fact, for open mappings of class C^1, we always have that

    ^(u) ⊆⋃_k=0^n-2_k(u);

in other words, u is a local homeomorphism near any point in _n-1(u) <cit.>. 
For example, the homeomorphism x↦ x^3 of the real line has a rank zero point at the origin.



    In the theory of parametric minimal surfaces, one considers weakly conformal parametrizations u^2→^m of a (possibly branched) minimal surface S=u(^2). Precisely, u must satisfy:
    
        
  *  weak conformality: |u_x|=|u_y| and u_x· u_y=0;
        
  *  minimality: Δ u = 0.
    
    If u satisfies (<ref>) then (u)=_0(u). Also, it is known that if, in addition, (<ref>) holds and u is minimizing, then (u) = ∅, under appropriate conditions on the boundary data, see <cit.> and the references therein. However, this result is not applicable in our setting, since minimizing constraint maps generally do not satisfy (<ref>), a simple example being given by (<ref>).


 We also note that, for harmonic maps between two-dimensional domains, a fine description of their behavior at the branch points was presented in <cit.>.






We now want to obtain some information on the sets _k(u), in particular concerning their dimension and rectifiability properties. For reasons that will become apparent later, we will focus on the case k=0. The next result follows from essentially standard considerations, cf. <cit.> for the case of harmonic maps. As we shall explain below, the following result is essentially contained in <cit.>.


Let M⊂^m be a smooth domain and let u∈ W^1,2(Ω;M) be a non-constant minimizing constraint map.  For any ball B such that 2B⋐Ω∖Σ(u),  we have the estimates
    
    ^n-2(B∩_0(u))   ≤ c ,
    ^n-2/2 (u(B∩_0(u)))   ≤ c,
 
for some constant c depending only on n, ∂ M, Du_L^∞(2B), and ∫_2B |u-(u)_2B|^2 dx.
 


Recall that, by definition, _0(u) is a closed set disjoint from Σ(u).
By the Euler–Lagrange system (<ref>), any minimizing constraint map is a solution of an elliptic system of the type

    Δ u^i = a_ij^α D_α u^j  in Ω,      a_ij^α∈ L^2(Ω)∩ L_^∞(Ω∖Σ(u)),

where the regularity of the coefficients follows from Theorem <ref>. In particular, as remarked in <cit.> (and as we formulate more precisely in Appendix <ref>), due to the diagonal structure of this system, the standard scalar methods concerning the critical set of elliptic equations are applicable. Therefore the measure estimate follows as in the scalar case <cit.>.  We remark here that the methods of <cit.> are not directly applicable, since minimizing constraint maps are not smooth near regular points; their methods require increased smoothness of the solution as the frequency increases. Nonetheless, the work <cit.> deals precisely with elliptic equations with rough coefficients.

Concerning estimate (<ref>), we argue as follows. 
By (<ref>),  for any δ>0 we can take points x_i∈_0(u) and radii r_i ∈ (0,δ), for i∈ I, such that 
    B∩_0(u)⊂⋃_i∈ I B_r_i(x_i),     ∑_i∈ I r_i^n-2≤ c.

Since u∈ C^1,1(B) and |Du| = 0 on B∩_0(u), we can find a uniform λ>0 such that u(B_r_i(x_i))⊂ B_λ r_i^2(u(x_i)), for every i∈ I.  Since {B_λ r_i^2(u(x_i)) : i ∈ I} is a covering of u(B∩_0(u)), letting δ→ 0 we get 

    ^n-2/2(u(B∩_0(u)))≤λ^n-2/2c.

as desired. 



Note that the estimate (<ref>) on the image of _0(u) is stronger than what one could deduce from Sard's Theorem whenever n ≤ 3 ≤ m or n > m, since minimizing constraint maps are only C^1,1.



For a smooth harmonic map u, Sampson <cit.> proved the following unique continuation result: if _1(u) has non-empty interior, then _1(u)= Ω and u maps into a geodesic. Then, it was shown in <cit.> that if u does not map into a geodesic, the set _1(u) has a locally finite ^n-1-measure.  
Perhaps surprisingly, the same unique continuation type properties do not hold in general[An obvious exception is for smooth harmonic maps into analytic manifolds, since in that case (Du) is constant in an open dense set.] for the sets _k(u) whenever k≥ 2, see e.g. <cit.>. 




 §.§ Branch points at free boundary points



We now turn our attention to the interaction between (u) and the free boundary ∂ u^-1(M). We first make the observation that, under rather general circumstances, one can make degree-theoretic arguments to produce branch points on the free boundary. Recall that  𝔹^n⊂^n denotes the open unit ball.


Let n≥ 3 and let u∈ W^1,2_g( B^n;^n\ B^n) be a minimizing constraint map. Assume that g∈ C^2( S^n-1; S^n-1) satisfies |(g)|≥ 2, and that, for a.e. >0 sufficiently small, the set  {|u|≥ 1+} is a topological annulus. Then 

    ⋃_k=0^n-2_k(u) ∩∂{|u|>1}≠∅.




From Theorem <ref> and Sard's theorem, we deduce that a.e. >0 sufficiently small is a regular value of |u|. Thus 

    D|u|= (u· Du)/|u|≠ 0    on  N_ := ∂{|u|>1+},

 and N_ is then a smooth manifold.  Since | (g)|≥ 2, the map g S^2→ S^2 is surjective. Thus, our assumption that the set {|u|≥ 1+} is a topological annulus for a.e. >0 sufficiently small, essentially amounts to saying that the manifold N_ is connected. 

Let us suppose, for the sake of contradiction, that 

    ⋃_k=0^n-2_k(u) ∩ N_0 = ∅.

We write, as usual, Π∘ u = u/|u|, so that u=|u|(Π∘ u) and 

    Du = (Π∘ u)⊗ D|u| + |u| D(Π∘ u).

Since  1≤ |u| is C^1,1 by Theorem <ref>, and attains a minimum on N_0, we have 

    D|u|=0   on N_0.

Notice also that, since Π is 0-homogeneous, we have

    u· D (Π∘ u )=0

and so D(Π∘ u) is always a singular matrix. Hence, 
combining (<ref>)–(<ref>),  we have 

    (D(Π∘ u))=n-1   on  N_0,

and by continuity the same also holds on N_, for >0 small enough.
Let us now choose  so that (<ref>) holds.  It follows from (<ref>) and (<ref>) that u· Du= |u| D|u|, therefore

    (Du)=n  on  N_.

We have seen in Proposition <ref> that 

    (u, B^n ∩{|u|>1},y) = (g)

whenever 1<|y|<λ. Thus, since N_={|u|=1+} is connected, taking |y|=1+ we have u^-1(y)⊂ N_ and so, according to (<ref>) and the definition (<ref>) of the local degree, we see that

    (u, B^n∩{|u|>1},y) = ∑_x∈ u^-1(y)( Du(x)) = (g).

 We now argue similarly to  the last step in the proof of Proposition <ref>. For x∈ N_, let ℬ_x:={τ^1_x,…,τ^n-1_x} be an oriented orthonormal basis for T_x N_. Thus D_τ^i |u|=0 on N_ and, by (<ref>), we have

    D_τ u(x) = Du(x)τ_x=D(Π∘ u)(x)τ_x.

Note that ν_x:=D|u|(x)/| D|u|(x)| is the unit normal to N_, which is well-defined by (<ref>), and so

    (D_ν|u|)(x) = (D|u|)(x) ·ν_x = | (D|u|)(x)|>0.

Hence, if we consider the map p_ N_→ S^n-1 defined by p_ := u/(1+), the last two identities show that, on N_, we have

    ( Du) = ( dp_),

where as usual dp_ TN_→ T S^n-1 is the differential between tangent spaces.  In particular, we deduce that

    (p_) = ∑_x: p_(x)=y/(1+)( dp_(x)) = (g).

Note that, by (<ref>), p_ N_→ S^n-1 is a local diffeomorphism. Since n≥ 3, S^n-1 is simply connected and so, since N_ is connected, p_ is actually a global diffeomorphism; in particular, we must have p_=1. However,  this contradicts (<ref>), since we assumed that |(g)|≥ 2.



Under the assumption that the non-coincidence set {|u|>1}, or a small perturbation of it, is a topological annulus, Proposition <ref> produces branch points of low rank on the free boundary.  This topological condition seems hard to verify in general, but we expect it to hold in any sufficiently symmetric scenario. 

 In any case, we are especially interested in knowing whether there are points with zero rank on the free boundary, and it seems that the existence of such points cannot follow from purely topological considerations, such as those used in the proof of Proposition <ref>.
Rank zero points are especially relevant to us since, at those points, (<ref>) becomes a degenerate obstacle problem, regardless of what obstacle M^c is; furthermore, for uniformly convex obstacles,  these are the only points at which (<ref>) is degenerate.  To construct such points, the proof of Proposition <ref> naturally leads us to consider axially symmetric maps: if u B^3→^3\ B^3 is a C^1 map which rotates at least twice about the z axis, then a simple calculation reveals that the gradient of u along the (x,y) plane must vanish on the z axis, cf. (<ref>).  In other words, the differential of the map p_ constructed in the proof of Proposition <ref> vanishes on the z-axis. The construction and complete analysis of points in _0(u)∩∂{|u|>1} in the axially symmetric case is actually quite involved, and it will occupy the final sections of this paper.

We will now show that the existence of rank zero points on the free boundaries is substantially simpler if we allow ourselves to consider non-convex obstacles. 
In fact, we can construct an artificial obstacle that touches the image of the maps u_k defined in (<ref>) over a cone with vertex at 0, cf. Figure <ref>. This example has the caveat that the obstacle is not convex, but on the other hand it has two advantages:

    
  * it shows that the bound _(_0(u) ∩∂ u^-1(M))≤ n-2, which follows from Proposition <ref>, is in general optimal;
    
  * it shows that branch points can lead to cone singularities on the free boundaries; such singularities do not occur in the classical obstacle problem <cit.>.

In the next statement, [β] denotes the greatest integer less than or equal to β. 

 






    For each odd integer k≥ 3 and each α∈(0,1/2), there is a C^[k/2],α-domain M⊂^3 with compact  complement such that the following holds: the analytic map u_k, defined in (<ref>), satisfies 
    u_k(^2)⊂M and  
 ∂ u^-1(M) is a cone with a true branch point at its vertex.




We work with polar coordinates z=re^i θ with (r,θ)∈ (0,∞)×[-π,π].
Take a small angle θ_k ∈ (0,π/8) such that cos(k/2θ_k)≥1/2, and fix  λ≥ 1 to be chosen later. Let 




    C   :={r e^i θ: r>0, |θ|∈ [0,θ_k]},
    
        E_1   :={r e^i θ: r>0, |θ|∈ (θ_k,π 2 + θ_k]},
    
        E_2   :={r e^i θ: r>0, |θ|∈ (π 2 + θ_k,3 π4]},
    
        F   :={r e^i θ: r>0, |θ|∈ (3π4,π]}.

To define the obstacle, we set

    (z):=
    - z^k/2   if  z∈ C,
    
    p_k(π(z))-λ(z,∂ C)^[k/2]+α   if  z∈ E_1∪ F,

where p_k(z) is the [k/2]-th order Taylor polynomial of z↦ - z^k/2 at the origin, and π^2→C is the nearest point projection onto the convex set C. Note that ∈ C_^[k/2],α(C∪ E_1∪ F), and  (z) = -λ|z|^[k/2]+α for z∈ F. Now, one may easily extend  to E_2 so that ∈ C_^[k/2],α(^2). Finally, we let M be the epigraph of φ, namely 
    M={(z,x_3)∈^2×: φ(z)< x_3}.

Note that the image of u_k corresponds to the union of the graphs of the functions z↦±| z^k/2|.  Selecting λ > 1 large, corresponding to the choice of α∈(0,1/2), we observe that ∂ M intersects the graph of -| z^k/2| precisely on C, and lies below it otherwise.  Thus, u(^2)⊂M with u^-1(∂ M)= C.  


In the next remark, we observe another interesting aspect of the example constructed in Proposition <ref>.


Consider any integer k≥ 5; this restriction ensures that M is at least C^2. Note that the map u_k-u_k(0)=u_k vanishes to order 2 at the origin, since we can compute, in polar coordinates,

    |u_k(re^iθ)|=r^2 √( 1+ r^2k-4cos^2(kθ)).

However, if k≥ 5 is odd, the distance ρ∘ u_k of u_k to the obstacle vanishes at least to order k-1+α >  4, since

    u_k(r e^i π/2) = -r^2(1,0,0)    ρ∘ u_k(r e^iπ/2) = λ  (r^2)^[k/2]+α= λ r^k-1+α.
 This shows that, although u_k-u_k(0) vanishes to a fixed order at the origin, ρ∘ u_k can vanish to arbitrarily high order there. 


The above example is only possible because the obstacle M^c in Figure <ref> is not uniformly convex. In fact, the next proposition, which builds on the results of Sections <ref> and <ref> and will be useful in Section <ref>, shows that, 
 for uniformly convex obstacles, ρ∘ u vanishes twice as fast as u-u(x_0) at a free boundary point x_0; by the above remark, this is false for general obstacles.



    Let M⊂^m be a smooth obstacle, and let u∈ W^1,2(Ω;M) be a weakly constraint map such that u ∈ C(B;(∂ M)) in a ball B⊂Ω. Then for every x_0∈ B∩∂ u^-1(M), there exists k∈ such that u - u(x_0) vanishes exactly to order k at x_0, and ρ∘ u vanishes at least to order 2k at x_0. In addition, if all the principal curvatures of ∂ M ∩ u(B) are uniformly positive, such as when M^c is uniformly convex, then ρ∘ u vanishes exactly to order 2k at x_0. 



Since u∈ C(B;(∂ M)), it follows from <cit.> that u∈ C_^1,1(B). By (<ref>), we can apply the classical results concerning the frequency of solutions to elliptic systems (cf. Appendix <ref>). 
This yields the existence of k ∈ℕ and a non-trivial (vector-valued) harmonic polynomial P_k of degree k, which is homogeneous around x_0, and a radius r_0 > 0 small (such that B_4r_0(x_0)⊂ B),  such that 

    D^j(u - u(x_0) - P_k)_L^∞(B_r(x_0))≤ c_σ r^k - j + σ,   j∈{0,1},

for every r∈(0,4r_0) and for each σ∈ (0,1), where c_σ > 1 is independent of r.
In particular u-u(x_0) vanishes to order k∈, proving the first assertion of this proposition. Moreover, replacing r_0 with a smaller radius if needed, we also obtain 
        
    N(u, x,r)≤3/2 k,

        for all r∈(0,r_0) and any x∈ B_r_0(x_0). Here N(u,x,r) is the frequency formula defined in (<ref>). 



We now turn our attention to ρ∘ u. 
Note that, as a consequence of (<ref>), we have

    Du _L^∞(B_r(x_0))≤ cr^k-1

for every r∈(0,r_0). 
Also, thanks to <cit.>, (<ref>) holds in B_4r_0(x_0). Therefore, by (<ref>), (<ref>), (<ref>), Corollary <ref>, and (<ref>), we get
    
    Δ(ρ∘ u)_L^∞(B_r(x_0))≤ cr^2k-2

    for every r∈ (0,r_0). As ρ∘ u≥ 0 in Ω, a standard blow-up argument (see <cit.>) yields a constant c_0, which may depend only n and k, such that 
    
    ρ∘ u _L^∞(B_r(x_0))≤ c_0cr^2k,

    for every r∈(0,r_0). This proves that ρ∘ u vanishes at least to order 2k at x_0. Moreover, it follows from (<ref>) that Π∘ u - u(x_0) vanishes at x_0 exactly to order k. Therefore, the first part of the proposition is proved. 


Assume now that all the principal curvatures of ∂ M∩ u(B) are uniformly positive.
Taking a smaller r_0 if necessary, we obtain a constant κ > 0 such that 

    ρ(ξ,ξ) ≥κ |ξ^⊤|^2  in M∩ u(B_4r_0(x_0)),

cf. (<ref>).
The rest of the proof is similar in spirit to that of Lemma <ref>, in that we will show that there is a small η > 0 such that if ρ∘ u < η r^2k in B_4r(x_0) for some r > 0, then ρ∘ u = 0 in B_r(x_0). Once this claim is proved, from our choice of x_0∈ B∩∂ u^-1(M) we deduce that 

    sup_B_r(x_0) (ρ∘ u) ≥η r^2k
 
for all r∈(0,r_0/4), as desired. 

For the moment, let us fix a scale r∈(0,r_0/4), and consider the rescaling

    ũ_x_0,r(x) := 1/r^k (u(x_0 + rx) - u(x_0)).

Note from (<ref>) that (<ref>) holds with |a_ij^α|≤ |Du| a.e. in Ω. Thus, by (<ref>), 

    Δũ_x_0,r = ã_ij^α D_αũ_x_0,r^j   a.e. in B_4,

where, by (<ref>),  all coefficients satisfy

    ã_ij^α_L^∞(B_4)≤1/r^k-1 a_ij^α_L^∞(B_4r(x_0))≤c/r^k-1Du_L^∞(B_4r(x_0))≤ c.

By (<ref>), we also have 

    N(ũ_x_0,r,0,4)  = N(u , x_0,4r) ≤3/2k.

Hence, Theorem <ref> yields 
        
    1/r^n - γ(k-1)∫_B_2r(x_0) |Du|^-γ dx = ∫_B_1 |Dũ_x_0,r|^-γ dx ≤ c,

        for some constants γ>0 and c> 1, independent of r. 
        
        We now argue as in the proof of Lemma <ref>. Fix a pair 1≤ t≤ T≤ 2. We observe from (<ref>) that analogously to (<ref>) we have 

    c/(T-t)^2r^2∫_B_Tr(x_0) (ρ∘ u)  dx≥ c_0 ∫_B_tr(x_0)∩ u^-1(M) |Du|^2 dx,

with some c>1 large and 0 < c_0 < 1 small now depending on κ above, but independent of s, t, and r. Using (<ref>) in place of (<ref>), we deduce from the generalized Hölder inequality, just as in (<ref>) and (<ref>), that 

    |B_tr(x_0)∩ u^-1(M)|^2/p≤c^2/γ/r^2k-2∫_B_Tr(x_0)∩ u^-1(M) |Du|^2 dx ≤c/(T-t)^2 r^2k∫_B_Tr(x_0) (ρ∘ u)  dx,

where p> 1 is chosen such that 1/p = 1/2 + 1/γ. Since the last inequality holds for all 1≤ t< T≤ 2, we can deduce from Lemma <ref> that

    sup_B_2r(x_0) (ρ∘ u) < η r^2k    ρ∘ u = 0  in B_r(x_0),

for some sufficiently small η > 0. Here the smallness parameter η can be chosen independently of r, as all constants appearing in (<ref>) are independent of r (nor of t and T). The implication of (<ref>) leads us to x_0∉B∩∂ u^-1(M), which gives us a contradiction. Thus, we must have 

    sup_B_r(x_0) (ρ∘ u) ≥η r^2k.

This proves that the vanishing order of ρ∘ u is at most 2k, provided that all principal curvatures of ∂ M∩ u(B) are uniformly positive, as desired. 








§ AXIALLY SYMMETRIC CONSTRAINT MAPS: REGULARITY THEORY




In this section, we initiate the study of Dirichlet energy minimizers in the class of axially symmetric maps that are constrained to take values outside the unit ball.

The regularity theory of such maps is parallel to that of minimizing constraint maps, which we developed in the last sections, but it is independent of it. Indeed, as we will see in Corollary <ref> below, minimizers in axially symmetric classes are not minimizing constraint maps in general. Nonetheless, in a neighborhood of any regular point, minimizers in axially symmetric classes are also minimizers in the full Sobolev space of admissible maps, cf. Appendix <ref>.



This section is divided into three parts. In Subsection <ref> we define the basic classes of maps under consideration. In Subsection <ref> we develop the partial regularity theory for minimizers in axially symmetric classes; the results of this subsection are the analogue of the classical results for minimizing constraint maps presented in Section <ref>. Finally, in Subsection <ref>, we prove the main results of this section, Theorems <ref> and <ref>, which are the analogue of Theorems <ref> and <ref>, respectively, in the axially symmetric setting. We emphasize that the proof of the L^-γ estimate in the axially symmetric setting is rather different from the one presented in Section <ref>, and we believe it is of independent interest.



 §.§ Setup and definitions


Throughout this section, we will use the following:


    Set e_3 := (0,0,1). We use cylindrical coordinates (r,θ,z) in the domain. We write 𝔻={(r,z): r^2 + z^2 < 1, r ≥ 0} for the half-disk in the (r,z)-plane. We shall write D' = (∂_r,∂_z) and define Δ' := ∂_rr + ∂_zz + 1/r∂_r, so that Δ= Δ' + 1/r^2∂_θθ  is the usual Laplacian in ^3. 


We now define axially symmetric maps, and the corresponding energy minimizers.



Fix k∈.

    
  *  A measurable map u  B^3→^3 is said to be k-axially symmetric if it is of the form
    
    u = ϱ (sinφcos kθ,sinφsin kθ,cosφ),

    for some pair (ϱ,φ):→ [0,∞)×[0,π].

  * A map u∈ W^1,2( B^3;^3\ B^3) is said to be energy minimizing among k-axially symmetric maps if u is itself k-axially symmetric and it satisfies 

    [u]≤[ũ]

for all k-axially symmetric maps ũ∈ W^1,2_u( B^3;^3\ B^3). 




    Note that (<ref>) implies 
    
    u∘ R_θ = R_kθ∘ u,

    for every θ∈[0,2π], where R_θ denotes the rotation about the z-axis by angle θ. However, (<ref>) is not sufficient to deduce the representation (<ref>). In fact, (<ref>) allows additional rotation of each horizontal circle in ^3 (i.e. a circle of form {r = r_0, z = z_0}) by an amount depending on (r_0,z_0). However, minimizers of the energy among all maps that verify (<ref>) subject to a k-axially symmetric boundary condition, in the sense of (<ref>), turn out to be k-axially symmetric as well.  We refer the reader to <cit.> for a similar argument in a slightly different context.


In what follows, given any pair of function (ϱ,)→ [1,∞)× [0,π], we define its energy through

    _k[ϱ,] := ∬_𝔻( |D'ϱ|^2 + ϱ^2( |D'φ|^2 + k^2/r^2sin^2)) r dr dz.

We call (ϱ,φ) admissible if in addition _k[ϱ,φ] < +∞. 


Note that u∈ W^1,2( B^3;^3∖ B^3) is k-axially symmetric if and only if the associated pair (ϱ,φ) is admissible, since 

    [u] = ∫_ B^3 |Du|^2 dx = 2π_k[ϱ,φ].

Thus u ∈ W^1,2( B^3;^3∖ B^3) is -minimizing among all k-axially symmetric maps if and only if the associated pair (ϱ,φ) is _k-minimizing among all admissible pairs,  i.e.

    _k[ϱ,φ] ≤_k[ϱ̃,φ̃]

for every admissible pair (ϱ̃,φ̃) satisfying (ϱ -ϱ̃)∪(φ -φ̃) ⋐. 

 



 §.§ Partial regularity theory



We now turn to the partial regularity theory for energy minimizers among k-axially symmetric maps.  Throughout this subsection we will always write u∈ W^1,2( B^3;^3\ B^3) for a fixed energy minimizer among k-axially symmetric maps.

Let us begin with the Euler–Lagrange system for the energy minimizers, and their regularity away from the z-axis, i.e. away from {r=0}.


Let u∈ W^1,2( B^3;^3∖ B^3) be an energy minimizer among k-axially symmetric maps.
Then u∈ C_^1,1( B^3∖{r=0}) and it is a weak solution of 
    
    -Δ u = |Du|^2 u χ_{|u|=1}  in  B^3.

Moreover, |u|-1 ∈ W^1,2( B^3) is nonnegative and weakly subharmonic. 





Throughout the proof we work on a fixed domain Ω⋐∖∂ = {(r,z): r^2 + z^2 < 1, r >0}. Note from Remark <ref> that (ϱ,φ) is a minimizer of the energy _k among all admissible pairs. With φ fixed, we see that ϱ minimizes _k[·,φ], i.e., _k[ϱ,φ] ≤_k[ϱ̃,φ] for every admissible pair (ϱ̃,φ) with (ϱ-ϱ̃) ⋐. Note that the coefficient of ϱ^2 in _k[ϱ,φ] is nonnegative and belongs to L^1(Ω), due to the constraint ϱ≥ 1. Let β∈ C^1((-∞,∞)) be such that  β = 0 in (-∞,1), β = 1 in [2,∞) and β' ≥ 0 in [0,∞). Then (ϱ + β((ϱ-1)/)ψ,φ) is an admissible pair, for all small > 0, for each ψ∈ C_c^1(Ω). Hence, the minimality of (ϱ,φ) implies that  

    Δ'ϱ≥ϱ(|D'φ|^2 + k^2/r^2sin^2φ)χ_{ϱ > 1}≥ 0   in Ω,

in the weak sense. Since the coefficients of Δ' are bounded away from zero and infinity in a neighborhood of Ω, the local L^∞-estimate <cit.> for weak subsolutions applies and so ϱ∈ L_^∞(Ω).


Let us now consider the variation of _k[ϱ,·] with ϱ fixed. Although an admissible φ̃ is constrained to taking values in [0,π], we can actually consider unconstrained variations  in deriving the Euler–Lagrange equation for φ. Indeed, given any φ̃→ satisfying _k[ϱ,φ̃] < ∞, the pair (ϱ,max{min{φ̃,π},0}) becomes admissible, and 
_k[ϱ, max{min{φ̃,π},0}] ≤_k[ϱ, φ̃].
Thus, the minimality of (ϱ,φ) implies _k[ϱ, φ] ≤_k[ϱ, φ + ψ] for any > 0 small, for each ψ∈ C_c^∞(Ω), from which we deduce that 

    D'· (rϱ^2 D'φ)  = k^2ϱ^2/2rsin 2  in Ω

in the weak sense. 
Note from our earlier discussion that ϱ∈ L_^∞(Ω). Moreover, since _k[ϱ,φ] < + ∞ and ϱ≥ 1 in , the right-hand side of (<ref>) belongs to L_^2(Ω). Thus, the higher integrability estimate <cit.> applies to (<ref>) and we see that ∈ W_^1,p(Ω) for some exponent p = p(Ω) > 2. Since Ω is a two-dimensional domain, the Sobolev embedding implies φ∈ C_^0,σ̅(Ω), with σ̅:= 1-2/p. 

Let us return to ϱ. Noting that (ϱ +ψ,φ) is an admissible pair for any >0 small, for each ψ∈ C_c^∞(Ω) nonnegative, we obtain from the minimality of (ϱ,φ) that  

    Δ'ϱ≤ϱ(|D'φ|^2 + k^2/r^2sin^2φ)  in Ω,

in the weak sense. Since φ∈ W_^1,p(Ω) with some p > 2 and ϱ∈ L_^∞(Ω), the right-hand side of (<ref>) belongs to L_^p/2(Ω). By (<ref>) and (<ref>), we obtain from <cit.> that ϱ∈ W_^2,p/2(Ω). 
Thus, since once again Ω is a two dimensional domain, the Sobolev embedding implies ϱ∈ C_^0,2σ̅(Ω) (recall σ̅= 1-2/p). Hence (ϱ +ψ,φ) is an admissible pair for any > 0 small and for each ψ∈ C_c^∞(Ω∩{ϱ > 1}), i.e. ψ can change sign in the region where ϱ>1. Thus, the minimality of (ϱ,φ) yields equality in (<ref>) in Ω∩{ϱ>1} in the weak sense. However, as  ϱ has weak second derivatives in Ω, we also have Δ'ϱ = 0 a.e. in Ω∩{ϱ = 0}. The last two observations combined yield  

    Δ'ϱ = ϱ(|D'φ|^2 + k^2/r^2sin^2φ)χ_{ϱ > 1}  a.e. in Ω.
 

We now return once again to .  Since ϱ∈ C_^0,2σ̅(Ω), we can apply the Hölder estimate for the first derivatives <cit.> to (<ref>) and deduce that φ∈ C_^1,2σ̅(Ω). This in turn yields that the right-hand side of (<ref>) belongs to L_^∞(Ω)∩ C_^0,2σ̅(Ω∖∂{ϱ>1}), so by the interior C^1,1-estimate <cit.> for the classical obstacle problem we obtain ϱ∈ C_^1,1(Ω).  Plugging this information into (<ref>), we deduce again from the higher-order regularity theory <cit.> that ∈ C_^2,σ(Ω) for every σ∈(0,1). In particular, the Euler–Lagrange equation (<ref>) for φ holds in Ω in the classical sense. 

Since Ω was an arbitrary domain compactly contained in ∖∂, the above observations show that (ϱ,φ) ∈ C_^1,1(∖∂)× C_^2,σ(∖∂). Thus, in light of (<ref>), we must have u∈ C_^1,1( B^3∖{r=0}). Moreover, as the Euler–Lagrange equations (<ref>) and (<ref>) hold a.e. in ∖∂, a direct computation yields that the Euler–Lagrange system (<ref>) holds in B^3∖{r=0} in the strong sense. Since u∈ W^1,2( B^3;^3) and the z-axis has zero hamronic capacity, we deduce that (<ref>) holds in the weak sense. Finally, the fact that |u| - 1 is weakly subharmonic in B^3 follows either directly from (<ref>) or from Lemma <ref>.



    It is an interesting problem to decide whether, in the setting of Lemma <ref>, we have φ∈ C^2,1_(𝔻), see <cit.> for the one-dimensional case and  <cit.> for the two-dimensional case.


Lemma <ref> shows that u is regular away from the z-axis. We now prove the following -regularity theorem on the z-axis, which is the analogue of Theorem <ref> and <cit.> in our setting:


    There are absolute constants _0 > 0 and c>1, such that if  B_4s(z_0e_3)⊂ B^3 and E(u,z_0e_3,2s) ≤_0^2, then u ∈ C^1,1(B_s(z_0e_3)) and 
    
    D^j u_L^∞(B_s(z_0e_3))≤c/s^j,   j=1,2.




The idea follows essentially from <cit.>. Translating and rescaling u in the domain, we can assume that z_0=0, s=1, and u is defined in 4 B^3. We first note that

    |D'(cos)| ≤|sin| (|∂_r φ|+ |∂_z |) ≤1/2(|D'|^2 + sin^2 /r^2) r

and so, since ϱ≥ 1 and k≥ 1, we have

    ∫_ D |D'(cos)|   d x ≤_k[ϱ, ] = E(u,0,1)/2π≤_0^2/2π.

By Fubini's Theorem and the fact that _k[ϱ,]<∞, for a.e. R>0 we have that |_S_R is continuous, where we write S_R = {(r,z)∈ D:r^2+z^2=R^2}.  Similarly, using (<ref>), we can further choose R so that

    _S_Rcos≤ c _0^2,

for a universal constant c. Since cos(0,[-1,1])⊆{0,π}, we can assume without loss of generality that (0,R)=(0,-R)=0, the case where (0,± R)=π being totally analogous. We thus see from (<ref>) that, if _0 is chosen small enough, then the image of B^3 under u lies in an upper half space, i.e.

    u( B^3)⊂ (^3\ B^3) ∩{x_3> 1 2}.

In other words, u( B^3) lies in a star-shaped region and so, by <cit.> (see also Appendix <ref> and in particular Theorem <ref> therein), we conclude that u∈ C^1,σ( B^3) for any σ∈ (0,1). The sharp regularity u∈ C^1,1( B^3) then follows from the standard theory for the scalar obstacle problem, as in <cit.>.


We now develop further properties of u. In analogy with Lemma <ref>, we have the following monotonicity formula:


     Let z_0∈ (-1,1) and 0<t<s < 1- |z_0| be given. Then 
     
    E(u,z_0e_3,t) ≤ E(u,z_0e_3,s),
 with equality if and only if u is 0-homogeneous about z_0e_3 in B_s(z_0e_3)∖ B_t(z_0e_3).



    Note that the minimizing property over the class of k-axially symmetric maps yields the stationarity over domain variations for balls centered on the z-axis. The rest of the proof is identical to the standard one, see e.g. <cit.>. 


As in Section <ref>, Lemma <ref> allows us to define the normalized energy density E(u,z_0 e_3, 0^+) through the limit E(u,z_0 e_3, r) as r↘ 0.

We now turn to the compactness properties of minimizers in the axially symmetric class, which is an analogue of Lemma <ref> in the axially symmetric setting.



    Let {u_i}_i=1^∞⊂ W^1,2( B^3;^3\ B^3) be a bounded sequence of energy minimizers among k-axially symmetric maps. Then there is a map u∈ W^1,2( B^3;^3\ B^3), which is energy minimizer among k-axially symmetric maps, such that u_i→ u strongly in W_^1,2( B^3;^3) along a subsequence.  


As in the classical case <cit.>, the key idea is to consider a suitable axially symmetric extension of the boundary values, see  <cit.> and also <cit.> in the setting of axially symmetric harmonic maps. Although the extension here gets slightly more involved due to the presence of the non-constant radial component ϱ, most of the argument is similar to the aforementioned literature. To maintain the technicalities to a minimum and keep the reading flow, we shall present the proof in Appendix <ref>. 



As a corollary of Lemma <ref> and Theorem <ref>, and using Lemmas  <ref> and <ref>, we can run the usual dimension reduction argument to deduce the following:


We have u∈ C^1,1_(Ω∖Σ(u)).    The singular set Σ(u) consists of discrete points on the z-axis, which are characterized by E(u,z_0 e_3, 0^+)≥_0^2.


From Theorem <ref> and Lemma <ref>, arguing just as in Section <ref>, we also obtain:


    The function |u|-1 is nonegative, subharmonic and  continuous on B^3, with a  universal modulus of continuity.   





 §.§ Regularity near free boundaries


We now turn to the regularity theory of energy minimizers among k-axially symmetric maps near their free boundaries. A crucial ingredient in our analysis is the uniqueness and structure of blow-ups in the axially symmetric class, see <cit.> and <cit.>.  In what follows,  we use the following standard terminology:


Given z_0∈ (-1,1), we call ϕ_z_0 a tangent map of u at z_0e_3 if it is a weak W_^1,2-limit of a sequence u_r_j,z_0e_3:= u(z_0e_3 + r_j·), where r_j→ 0 as j→∞. 

 Combining Lemmas <ref> and <ref> with Corollary <ref>,  it follows that z_0e∈Σ(u) if and only if there is a non-constant tangent map ϕ_z_0 of u at z_0e, which is necessarily 0-homogeneous and minimizing among k-axially symmetric maps. In our setting, there are only four different non-constant tangent maps:



Let u∈ W^1,2( B^3;^3\ B^3) be an energy minimizer among k-axially symmetric maps. 
For every point z_0e_3∈Σ(u), there exists a unique tangent map ϕ_z_0 which, when restricted to 𝕊^2, is one of the four maps 

    ± S^-1∘ψ_k ∘ S    or   ± S^-1∘ψ_k∘ S,

where ψ_k→ is the map w↦ w^k, and S𝕊^2→ is the stereographic projection from e_3.
Moreover, there is a modulus of continuity ω, depending only on k, (z_0e_3,∂Ω), and u_W^1,2(Ω), such that 

    |u(x) - ϕ_z_0 (x- z_0e_3)| ≤ω(|x-z_0e_3|),

for every x∈Ω with |x-z_0e_3|≤1/2(z_0e_3,∂Ω). 



According to Corollaries <ref> and <ref>,  we have Σ(u)⊂{|u|=1}∩{r=0}. Thus, by Lemmas <ref> and (<ref>), at any point in Σ(u), all tangent maps of u at that point are 0-homogeneous harmonic maps into S^2, which further minimize the energy among k-axially symmetric harmonic maps. The first claim now follows from the classification of such maps from <cit.>. The second claim, concerning the existence of a modulus of continuity, is a direct consequence of the compactness of minimizing maps provided by Lemma <ref>,  and the fact that there is only a finite number of tangent maps. Indeed, since the function r↦ u_r,z_0 e_3 is continuous into W^1,2( B^3), the closest tangent map to u_r,z_0 e_3 varies continuously with r. But since there is only a finite number of such maps, such a map must eventually always be the same for r sufficiently small.



Let k≥ 2 and let u∈ W^1,2( B^3;^3∖ B^3) be an energy minimizer among k-axially symmetric maps. If z_0 e_3 ∈Σ(u) then u is not energy minimizing in any neighborhood of z_0 e_3.



    As described before the proof of Proposition <ref>, to  any point z_0e_3∈Σ(u) one can associate tangent maps ϕ_z_0 of u at z_0e_3, which in particular are non-constant 0-homogeneous k-axially symmetric maps. If u was minimizing near z_0 e_3 (in the full Sobolev space of admissible maps), then according to Lemma <ref> so would ϕ_z_0. However, by k-axial symmetry it is easy to see that 
    ϕ_z_0 S^2→ S^2 has (ϕ_z_0)=k. Thus, according to <cit.>, ϕ_z_0 is not a minimizer.




We can now prove the first main result of this section, which is an extension of Theorem <ref> to the setting of energy minimizers among k-axially symmetric maps.




Let u∈ W^1,2( B^3;^3\ B^3) be energy minimizing among k-axially symmetric maps. There are positive constants s,γ, and c, which depend only on k and u_W^1,2( B^3), such that if z_0e_3∈Σ(u) for some z_0∈ (-1,1), then 
    
    ∫_B_s(z_0e_3) |Du|^-γ dx ≤ c.





Let us write Λ:=u _W^1,2( B^3) for simplicity. By pre-composing u with a translation along the z-axis, which of course preserves the k-axial symmetry, we can assume that z_0=0.
Furthermore, by Lemma <ref> and Corollary <ref>, we deduce that B_s̅∩Σ(u) = {0}, for some s̅ =s̅(k,Λ). 

Consider the 0-homogeneous rescalings u_s(y) := u(sy), and let ϕ_0 be as in Proposition <ref>. This proposition also yields a modulus of continuity ω, depending only on k and Λ, such that u_s - ϕ_0_L^∞( S^2)≤ω(s) on S^2 for all s < 1/2. Since ϕ_0|_ S^2 is smooth and u_s|_ S^2 is uniformly close to it, ε-regularity implies that it must also be continuous and therefore C^1,1. Hence, given α∈ (0,1),  by interpolation we can find a new modulus of continuity ω, which still depends only on k and Λ, such that

    u_s -ϕ_0_C^1,α( S^2)≤ω(s).
 
 In light of Proposition <ref>, we can find ,δ > 0, depending only on k, such that

    |Dϕ_0| ≥ 2δ  on  S^2∖ B_(± e_3).

Hence, combining the previous two estimates, we can take s̅ smaller if necessary, without changing its dependence on k and Λ, in such a way that 

    |Du_s| ≥δ  on  S^2∖ B_(± e_3)

for every s∈(0,s̅). Thus, taking the cone _ := {0}∪ (⋃_s > 0 B_s (± se_3)),

    ∫_B_s̅∖_ |Du|^-1 dx = ∫_0^s̅ s^2  ds ∫_ S^2∖ B_(± e_3) |Du_s|^-1 d^2 ≤ cδ^-1.

Therefore, by Jensen's inequality, it suffices to find some γ = γ(k,Λ) ∈ (0,1) such that 

    ∫_B_s̅∩_ |Du|^-γ dx ≤ c.

We now follow the arguments in Section <ref>. We first claim, analogously to Lemma <ref>, that

    sup_1≤ t≤ 4 N(u_s, ± te_3, 1) ≤λ_0,

uniformly for all s ∈ (0,2s̅), for some λ_0=λ_0(k,Λ), with possibly smaller s̅ = s̅(k,Λ). This claim can be verified by following the lines of the proof of Lemma <ref> without major modifications. We only need to employ Lemmas <ref>, <ref>, and Corollary <ref>  in place of Lemmas <ref>, <ref>, and Corollary <ref>, respectively. Note also that |u_s - ϕ_0| ≤ω(5s) in B_5 and |ϕ_0| = 1 replace Corollary <ref> in the proof; alternatively, this could also be inferred from Corollary <ref>. Apart from these changes the proof goes through unchanged, and so we shall not repeat the details here. 


Let us fix an absolute constant τ∈ [1,4) to be determined later. In light of Proposition <ref>, we have 

    |Dϕ_0| ≤ c   in B_4τ(± te_3)

uniformly for t∈[1,4].  Choosing >0 smaller if necessary, we also have 

    Du_s → Dϕ_0  uniformly in B_4τ(± te_3)

as s→ 0; here again the rate of convergence depends only on k and Λ. Thus, 

    |Du_s| ≤ 2c  in B_2τ(± te_3),

uniformly for all s∈(0,2s̅), with a possibly smaller s̅ = s̅(k,Λ). Thus, in view of (<ref>) and (<ref>), we can also employ Lemma <ref> to find δ,σ∈ (0,1), which depend only on k and Λ, such that 

    |{ |Du_s| ≥δDu_s_L^∞(3Q)}∩ Q|≥ (1-σ)|Q|,

for every cube Q with 3Q⊂ B_2τ(± t e_3). With (<ref>) at hand, we can proceed with the Calderón-Zygmund cube decomposition argument, just as in the proof of Theorem <ref>, to get that 

    ∫_B_τ(± te_3) |Du_s|^-γ dx ≤ c,

uniformly for all s∈(0,2s̅) and 1≤ t≤ 4. 

To conclude, we choose τ≥ 1 such that (B_4∖ B_1)∩_⊂∪_1≤ t≤ 4 B_τ(± te_3). By (<ref>), 

    ∫_(B_4∖ B_1)∩_ |Du_s|^-γ dx ≤ c,

uniformly for s∈ (0,2s̅). Choose ℓ̅ such that 2^-ℓ̅≥s̅ > 2^-ℓ̅-1. As s̅ depends only on k and Λ, so does ℓ̅. Then by (<ref>), 

    ∫_B_s̅∩_ |Du|^-γ dx    ≤∑_ℓ = ℓ̅^∞∫_(B_2^-ℓ+2∖ B_2^-ℓ)∩_ |Du|^-γ dx ≤∑_ℓ =ℓ̅^∞ 2^-ℓ(n + γ) c ≤ c.

This finishes the proof. 


Based on Theorem <ref>, and as shown  in Section <ref>, we can establish that energy minimizers among k-axially symmetric maps are regular around their free boundaries.


    Let u ∈ W^1,2( B^3;^3\ B^3) be an energy minimizer among k-axially symmetric maps. Then for every x_0 ∈ B^3 lying in the closure of {|u|>1}, u ∈ C^1,1(B_δ(x_0)) with the estimate 
    
    u _C^1,1(B_δ(x_0))≤ c,

    where δ > 0 and c>1 depend only on 1-|x_0|, k and u_W^1,2( B^3). 


 This result follows from Theorem <ref> by using the properties of energy minimizers in the axially symmetric class established in Subsection <ref>,  exactly as in the proof of Theorem <ref>: since the argument is identical, we omit it.





§ AXIALLY SYMMETRIC CONSTRAINT MAPS: BRANCH POINTS ON FREE BOUNDARIES


The goal of this section is to prove Theorem <ref>, concerning the structure of the free boundary points in _0(u), for an energy minimizer u∈ W^1,2( B^3;^3\ B^3) in the class of k-axially symmetric maps. 
Uunless otherwise stated, in what follows we assume that u is such a map and that
k≥ 2. As in (<ref>), we then write

    u=ϱ (sinφcos kθ,sinsin k θ, cos),

for some pair (ϱ,φ) D→[0,∞)× [0,π], with ϱ = ϱ (r, z) and φ = φ (r,z). We will also consistently use Notation  <ref> and,  to keep the notation as simple as possible, we will write v  B^3→^2 for the projection map of u onto the sphere ^2. Specifically,  v=Π∘ u, in the notation of Section <ref>:   

    v  := Π∘ u = u/ϱ = (sinφcos (kθ),sinφsin(kθ),cosφ)  in  B^3.

In particular, |Dv| is constant in θ, and 

    |Dv|^2 = (∂_rφ)^2 + (∂_zφ)^2 + k^2/r^2sin^2φ  in  B^3∖{r = 0}.

As mentioned in Section <ref>, our motivation for considering maps u as described above is that any free boundary point on the z-axis is necessarily a branch point in _0(u). Indeed, by Theorem <ref>, u is C^1,1 around the free boundary,  and thus Dϱ=0 there, since ϱ attains its minimum on the free boundary. On the other hand,  we have Dv=0 at any regular point on the z-axis. Indeed,  using the representation (<ref>), we see that, for a fixed z, the map f→ defined by f(w)=f(r e^iθ) :=(v^1+i v^2)(r,θ,z) satisfies

    ∂_w f = (cosφ ∂_r φ+k/rsinφ)e^i (k-1)θ,  ∂_w̅ f = (cosφ ∂_r φ - k/rsinφ)e^i (k+1)θ.

Since k≥ 2, both exponentials e^i(k∓ 1)θ cannot be extended continuously to  w=0 and so, at any point where v  (hence f)  is C^1, their coefficients in (<ref>) must converge to 0 as |w| = r→ 0^+. Since |cosφ(0,z)|=1 and sinφ(0,z)=0, we conclude that Dv vanishes at all regular points on the z-axis. Thus

    ∂{|u|>1}∩{r=0}⊆_0(u).

The results of this section can be seen as a much more precise version of the above calculation.  Indeed, we will perform a fine blow-up analysis of u near a free boundary point on the z-axis. The structure of u at such a point is quite rich and the next theorem, which is the main result of this section, summarizes the main conclusions of our analysis.



 Given k ≥ 2, let  u∈ W^1,2_g( B^3;^3\ B^3) be an energy minimizer in the class of k-axially symmetric maps. Set ϱ=|u| and v = u/|u| as in (<ref>), and assume that g∈ C^2( S^2;λ S^2) for some λ>1, and that g is k-axially symmetric with |(g)|=k.

Let z_0 e_3∈∂{ϱ>1} be an arbitrary free boundary point on the z-axis. Then:

    
  *  
    v∓ e_3 vanishes at z_0e_3 exactly to order k, and there are constants C,a,σ,s > 0 such that 
    
    | D^j( v - (ar^kcos k θ,ar^ksin kθ,± e_3))| ≤ C r^k - j +σ,    j = 0,1,2,

    whenever r^2 + (z-z_0)^2 < s^2;
    
  *  ϱ-1 vanishes exactly to order 2k at z_0e_3; also, if we define the sequence of rescalings
    
    ϱ_s(r,z):=ϱ(s r,z_0+s z)-1/a s^2k,

    then, for every sequence s_i→ 0^+, up to a subsequence there is a blow-up ϱ̃ such that ϱ_s_i→ϱ̃ in  W_^2,q(^3) for any q∈ [1,∞). Moreover, ρ̃ is a non-constant, k-axially symmetric, (2k)-homogeneous, non-negative solution of
    
    Δϱ̃= k^2 r^2k-2χ_{ϱ̃>0}  in ^3;

    
  *  
    there exists a modulus of continuity ω and a small constant s_0 > 0 such that 

    _(B_s(z_0e_3)∩∂{ϱ > 1}, {r = 0}∪ (Z_2k-1∖{z= 0})) < sω(s),

for all s ∈ (0,s_0), where Z_ℓ is the nodal set of the zonal harmonics of degree ℓ. In particular, the free boundary ∂{ϱ>1} cannot be represented as a C^1-graph near any point of _0(u).



Before proceeding with the blow-up analysis required to prove Theorem <ref>, let us show how it easily implies Theorem <ref>.


Let k≥ 2, λ > 1, and consider a k-axially symmetric map g ∈ C^2(^2; ^2) with |(g)|= k. Let u∈ W^1,2( B^3;^3∖ B^3) be energy minimizing among all k-axially symmetric maps in W^1,2_λ g( B^3;^3∖ B^3). Note that, by axial symmetry, the image of the z-axis under u is contained in the z-axis; on the other hand, the image of the z-axis under u is also disconnected, due to the constraint. Hence,  u is necessarily discontinuous, i.e. Σ(u)≠∅. 

By Corollary <ref>, Σ(u) is a discrete set on the z-axis. Fix any z_*e_3 ∈Σ(u). By Theorem <ref>, we can find a ball B_* centered at z_*e_3 such that B_*⊂{ϱ = 1}, where ϱ is as in (<ref>). Hence, we can consider the largest open and connected set U⊂{ϱ = 1} containing z_*e_3. Choose z_0e_3∈∂ U. Then z_0e_3∈∂{ϱ > 1} by the definition of U. By Theorem <ref> again, we can find a small ball B centered at z_0e_3 such that u ∈ C^1,1(B). 
Taking the radius of B smaller if necessary, we deduce from Corollary <ref> that u is the unique minimizing constraint map in W^1,2_u(B;^3∖ B^3). 

We now verify the claims in Theorem <ref>; note that the boundary condition |(g)|=k guarantees that u is necessarily non-constant. Claim (<ref>) follows from the choice of z_0e_3∈∂ U, which readily implies z_0e_3∈∂( int{ϱ = 1}) ≠∅. Claim
(<ref>) follows from Theorem <ref>(<ref>)-(<ref>). Finally, claim 
(<ref>) follows from Theorem <ref>(<ref>).


The proof of Theorem <ref> is rather involved and is divided over the next three subsections. 



 §.§ Proof of Theorem <ref>(<ref>)


We begin our analysis with a simple consequence of Lemma <ref>.


We have ϱ∈ C_^1,1( B^3∖Σ(u)), v∈ C_^2,σ( B^3∖Σ(u);𝕊^2), and φ∈ C_^2,σ( B^3∖Σ(u)) for every σ∈(0,1). Also, the pair (ϱ,v) solves 

    Δϱ = ϱ |Dv|^2χ_{ϱ>1}
    Δ v + 2/ϱ Dϱ· Dv = -|Dv|^2 v
      in  B^3,

in the strong sense. 



Note that (<ref>) follows directly from (<ref>); in fact, the computation works without k-axial symmetry. Concerning the regularity statements, let z_0e_3∈ B^3∖Σ(u). By Lemma <ref>, there is a ball B⊂ B^3 centered at z_0e_3 such that u∈ C^1,1(B). Since ϱ = |u|≥ 1 in B^3, we also have ϱ∈ C^1,1(B). We then have v∈ C_^2,σ(B) for every σ∈(0,1): this can be deduced from the second equation in (<ref>), cf. <cit.>. 

Finally, we deal with the regularity of φ. Writing v=(v^1,v^2,v^3) ∈ C_^2,σ(B;^3), we see that φ = arccos v^3 in B^3. Then, since arccos is continuous in [-1,1], and v^3 ∈ C^2,σ(B)⊂ C(B), we have φ∈ C(B). Since φ({r = 0}∩ B) ⊆{0,π}, shrinking the radius of B if necessary, we may assume without loss of generality that φ∈ [0,π/6]; the other case is symmetric by replacing φ with π-φ. Then we may also write φ = arcsin (v^1/(cos k θ)) in B∩{|θ|<π/2}. As arcsin is analytic in (-1,1), |v^1|≤sinφ≤ 1/2 in B, and v^1 ∈ C^2,σ(B), then φ∈ C^2,σ(B∩{|θ|<π/2) for every σ∈(0,1). But  does not depend on θ, so this last restriction on θ can be removed, completing the proof. 



Assume that z_0e_3∈ B^3∖Σ(u). By Lemma <ref> we have, for some s small, that ϱ∈ C^1,1(B_s(z_0e_3)) and v∈ C^2,σ(B_s(z_0e_3)) for every σ∈(0,1). Since ϱ≥ 1 in B^3 by the constraint,it follows that (1/ϱ)Dϱ∈ C^0,1(B_s(z_0e_3)). Moreover, by Lemma <ref>, 
|Dv|^2 ∈ C^1,σ(B_s(z_0e_3)) for every σ∈(0,1). 

Let ṽ B_s(z_0e_3) →^2 be the map consisting of the first two components of v, i.e., 

    ṽ := ( sinφcos kθ,sinφsin kθ).

Since the boundary datum g S^2→λ S^2 satisfies |(g) |=k, we see from Proposition <ref> that u is surjective onto the annulus λ B^3∖ B^3. In particular, we deduce that sin is not identically zero in B_s(z_0 e_3), since  satisfies the unique continuation property by (<ref>), and hence ṽ is also not identically zero in B_s(z_0 e_3).
We observe that each component of ṽ is a C^2,σ-solution of the second line of (<ref>), 
where we think of (1/ϱ)Dϱ and |Dv|^2 as coefficients of the equation. Therefore, it follows from <cit.> that each component of ṽ vanishes to finite order. Applying <cit.>  to each component of ṽ, we see that there is a non-trivial homogeneous harmonic polynomial P = (P^1,P^2) of degree d such that  

    |D^j (ṽ - P)| ≤ C (r^2 + (z-z_0)^2)^d-j + σ/2,   j∈{0,1,2},

whenever r^2 + (z-z_0)^2 < s^2; here we take s smaller if necessary, and the constant C may depend on σ but not on (r,z-z_0). 

Since P is homogeneous of degree d, (<ref>) implies that 

    P  = (pcos kθ,psin kθ),

where p is a homogeneous polynomial of degree d in (r,z). Moreover, since P is non-trivial, so is p. In addition, being φ a polar angle of v, it must satisfy sinφ≥ 0 in B^3. So, (<ref>), (<ref>), and (<ref>) altogether imply 

    p ≥ 0   in {r≥ 0}.
 
Then, the structure of homogeneous harmonic polynomials <cit.> shows that we must have d = k, and 

p (r,z) = ar^k,

for some a > 0; in other words, P is a zonal harmonic. Noting that ṽ = P = 0 on {r = 0}, it follows from (<ref>) that 

    |D^j (ṽ - P)| ≤ C r^k-j + σ,   j∈{0,1,2},

whenever r^2 + (z-z_0)^2 < s^2. 

Choosing s smaller if necessary, we can also assume that sin^2φ∈ [0,1/2] in B_s(z_0e_3). Thus, (<ref>) and (<ref>) together yield

    (∂_rφ)^2 + (∂_zφ)^2 + k^2/r^2sin^2φ≤ 2|Dṽ|^2 ≤ Cr^2k-2,

whenever r^2 + (z-z_0)^2 < s^2 and r>0. Also, since v(z_0e_3) = ± e_3, and sin^2φ∈ [0,1/2] in B_s(z_0e_3), we have

    |v^3 ∓ 1| ≤ Csin^2φ,    |Dv^3| ≤sinφ|Dφ|,  and   |D^2v^3| ≤ |D|^2 + sin^2φ |D^2φ|^2    in B_s(z_0e_3).

Therefore, (<ref>) along with φ∈ C^2,σ(B_s(z_0e_3)) yield

    |D^j (v^3 ∓ 1)| ≤ Cr^2k - j,    j =0,1,2,

whenever r^2 + (z-z_0)^2 < s^2. Combining (<ref>) with (<ref>), we arrive at (<ref>).




 §.§ Proof of Theorem <ref>(<ref>)


We begin by noting that, by Theorem <ref>(<ref>), the (2k-2)-homogeneous rescaling of the function |Dv|^2, which depends only on (r,z) as seen in (<ref>), converges to k^2a^2 r^2k-2, which is again (2k-2)-homogeneous. The following corollary of Theorem <ref>(<ref>) shows that this convergence occurs at an almost-Lipschitz rate. 


    Given z_0e_3∈ B^3∖Σ(u) and σ∈(0,1), there exist C,s_0 > 0  such that 
    
    |( r∂_r + (z-z_0)∂_z  - (2k-2))|Dv|^2 | ≤ Cr^2k-2 s^σ,

    whenever r^2 + (z-z_0)^2 ≤ s^2 and s∈(0,s_0). 


 
    By (<ref>), we have
    
    | D^j( |Dv|^2 - k^2 a^2 r^2k-2) | ≤ C r^2k - j -2 +σ,    j = 0,1,

    whenever r^2 + (z-z_0)^2 < s^2. Since (r∂_r + (z-z_0)∂_z) r^2k-2 = (2k-2)r^2k-2, the conclusion follows from (<ref>) and the triangle inequality. 


We now observe that, due to (<ref>), the function ϱ - 1, which corresponds to the distance map from B^3 to ^2, must vanish exactly to order 2k at every free boundary point. 


    Given z_0e_3∈∂{ϱ>1}, ϱ - 1 vanishes at z_0e_3 exactly with order 2k, and there exist s_0 > 0 small and C>1 large such that
    
    1/C s^2k≤sup_B_s(z_0e_3)(ϱ - 1 + s|Dϱ|) ≤ Cs^2k,

    for all s ∈ (0,s_0).  



    By Lemma <ref>, u∈ W^1,2( B^3;^3∖ B^3) is a weakly constraint map. Taking a small s_0 we have u∈ C^1,1(B_s_0(z_0e_3)). The conclusion then follows directly from Proposition <ref>.


Given any z_0∈ (-1,1), let us define the following Weiss-type <cit.> energy  

    W(z_0,s) := 1/s^4k+1∫_B_s(z_0e_3)(|Dϱ|^2+ 2(ϱ-1)ϱ|Dv|^2) dx - 2k/s^4k+2∫_∂ B_s(z_0e_3) (ϱ - 1)^2  d^2;

recall from (<ref>) that |Dv| is a function of (r,z).  In the scalar obstacle problem, the analogous energy is monotone. Here we have the following similar result: 


    Given any z_0e_3∈∂{ϱ>1} and every σ∈(0,1), there is s_0 > 0 such that s↦ W(z_0,s) + Cs^σ is non-decreasing in s ∈ (0,s_0), where C>1 may depend on s_0 and σ. In particular, W(z_0,0^+):=lim_s→ 0 W(z_0,s) exists. 



By Theorem <ref>, we have z_0e_3∉Σ(u). Let s_0 be a small radius to be determined and note that, from Lemma <ref>, d/dsW(z_0,s) exists for every s∈ (0, s_0).

Let a > 0 be as in (<ref>). In what follows, we fix s∈(0,s_0). For every s > 0 small, consider 

    ϱ_s (r,z) := ϱ(sr,z_0 + sz) - 1/a^2s^2k,   F_s (r,z) := |Dv|^2(sr,z_0 + sz) /a^2s^2k-2.

In view of (<ref>), ϱ_s solves  

    Δϱ_s = (1 + s^2kϱ_s)F_sχ_{ϱ_s>0}  in B_1/s_0,

in the strong sense. Let us also write ϱ̇_s = d/dsϱ_s and Ḟ_s = d/dsF_s, so that

    ϱ̇_s  = (r∂_r + z∂_z- 2k)ϱ_s/s,  Ḟ_s = (r∂_r + z∂_z- (2k-2))F_s/s.

Note that, thanks to Corollary <ref> and Lemma <ref>,

    ϱ_s + s|ϱ̇_s| +  F_s + s^1-σ |Ḟ_s| ≤C/a^2  in  B^3,

for every s∈(0,s_0). Also, we can rewrite 

    1/a^4W(z_0,s)     = ∫_ B^3( |Dϱ_s|^2 + 2ϱ_s(1 + s^2kϱ_s)F_s)  dx  - 2k ∫_^2ϱ_s^2 d^2.

Differentiating both sides of (<ref>) in s, integrating by part, and using (<ref>) for ϱ_s, we compute  

    1/a^4d/dsW(z_0,s)    = 2∫_ B^3(Dϱ_s· Dϱ̇_s + ϱ̇_s (1 + s^2kϱ_s) F_s)  dx - 4 k ∫_^2ϱ_sϱ̇_s d^2 
          + 2∫_ B^3ϱ_sd/ds( (1 + s^2kϱ_s)F_s) dx 
        = 2∫_^2 (r∂_rϱ_s + z∂_zϱ_s - 2kϱ_s)ϱ̇_s  d^2 
          + 2 ∫_ B^3ϱ_s ((1+ s^2kϱ_s)Ḟ_s + s^2k-1(2kϱ_s + sϱ̇_s)F_s) dx
       ≥  2 ∫_^2 sϱ̇_s^2  d^2 - C/a^4 s^σ - 1≥ - C/a^4 s^σ - 1,

where, in the second last inequality, we used (<ref>) and (<ref>). By (<ref>), we obtain 

    d/ds (W(z_0,s) + Cs^σ) ≥ 0,

for every s∈(0,s_0), which finishes the proof. 



Let s_i→ 0^+ be given 

and define ϱ̃_i := ϱ_s_i, and F̃_i := F_s_i, where ϱ_s and F_s are as in (<ref>). By Theorem <ref>(<ref>), we have 

    F̃_i → k^2r^2k-2  in C_^1,σ(^3).

In addition, due to Theorem <ref>(<ref>) and Lemma <ref>, a standard compactness argument yields, up to a non-relabeled subsequence,

    ϱ̃_i →ϱ̃  in C_^1,σ∩ W_^2,q(^3),

for every σ∈(0,1) and q∈[1,∞), for some non-trivial ϱ̃∈ C_^1,1(^3). Since ϱ̃_i is axially symmetric and nonnegative, so is its limit ϱ̃. The fact that ϱ̃ satisfies 
 (<ref>) follows from (<ref>) and (<ref>), along with the regularity ϱ̃∈ C_^1,1(^3). 

Thus, it remains to prove the (2k)-homogeneity of ϱ̃, and this follows from the almost monotonicity of the Weiss-type energy, as we now detail. By Lemma <ref>, the limit W(z_0,0^+) exists. However, analogously to (<ref>), we also have, for each t > 0, that

    1/a^4 W(z_0,s_it) = 1/t^4k+1∫_B_t( |Dϱ̃_i|^2 + 2ϱ̃_i(1+(s_it)^2kϱ̃_i)F̃_i) dx - 2k/t^4k+2∫_∂ B_tϱ̃_i^2 d^2.

Thus, passing to the limit in (<ref>) along with (<ref>) and (<ref>), we deduce from Lemma <ref> that 

    1/a^4W(z_0,0^+) = lim_j→∞ W(z_0,s_it) = W̃(t),

for all t > 0, where the energy W̃ is defined as 

    W̃(t) := 1/t^4k+1∫_B_t (|Dϱ̃|^2 + 2ϱ̃k^2r^2k-2) dx - 2k/t^4k+2∫_∂ B_tϱ̃^2 d^2.

Using  (<ref>), one can following essentially verbatim the proof of Lemma <ref> to show that 

    d/dtW̃(t) = 2 ∫_^2 tϱ̇̃̇_t^2  d^2,    ϱ̃_t(r,z)=ϱ̃(tr,tz)/t^2k,

cp. (<ref>). Since (<ref>) implies that W̃(t) is constant, we deduce that ϱ̃ is (2k)-homogeneous, as desired.





 §.§ Proof of Theorem <ref>(<ref>)

To complete our analysis, it remains to analyze the free boundary for ϱ̃. According to Theorem <ref>(<ref>),  ϱ̃ satisfies (<ref>), and a crucial feature of this equation is that the forcing term on the right-hand side is independent of z. In particular, ∂_z ϱ̃ is a (2k-1)-homogeneous k-axially symmetric harmonic polynomial on each component of {ϱ̃>0}. In other words, ∂_zϱ̃ is a zonal harmonic of degree (2k-1) on each connected component of {ϱ̃>0}. 

Recall that the zonal harmonics of degree ℓ are constant multiples of

    x↦ |x|^ℓ P_ℓ(cos),

where P_ℓ is the Legendre polynomial of degree ℓ.  The polynomial t↦ P_ℓ(t) has ℓ distinct zeros  in (-1,1), arranged symmetrically about t = 0, and the nodal set of P_ℓ is given by 

    Z_ℓ={P_ℓ(cos)=0}.
 We are of course specially interested in the case where ℓ=2k-1 is odd: in this case, the intersection of  Z_ℓ with S^2 is  the union of ℓ-latitude circles. We refer the interested reader to <cit.> for further details.

The previous discussion leads us to consider the connected components of ^3\ Z_ℓ.  Thus, for -1 ≤ t_1 < t_2 ≤ 1,  consider the axially symmetric region 

    _t_1,t_2 := {t_1 √(r^2 +z^2)< z < t_2 √(r^2 + z^2)},
 

and an axially symmetric 2k-homogeneous solutions q to 

    Δ q = k^2 r^2k-2  in _t_1,t_2.

By the axial symmetry of q,  we see that 

    q(r,z) = (r^2 + z^2)^k y(z/√(r^2 +z^2)),

where y(t_1,t_2)→ is a solution to 

    (1 - t^2) ÿ - 2t ẏ + 2k(2k+1) y = k^2 (1-t^2)^k-1,   t_1 <t < t_2.

The following result gives a description of the solutions to this ODE.


  Let y(t) be a solution of (<ref>) and assume that k≥ 2.  Then
  
    y  = c_1 P_2k + c_2 Q_2k + p_k,

where P_2k is the Legendre polynomial of degree 2k, Q_2k is the Legendre function of the second kind of degree 2k, and p_k is the particular solution to (<ref>) given by

    p_k(t) = ∑_i = 0^k a_2it^2i,

under the recurrence relation

    a_0=0,      a_2i + 2 = -(2k-2i)(2k + 2i + 1)a_2i + (-1)^i k^2 k-1i/(2i+2)(2i+1),  i = 0,1,2,⋯.

Moreover ṗ_k(t) ≠ 0 for every t∈ (-1,1)∖{0} where p_k(t) = 0. 



The claims concerning (<ref>)–(<ref>) are elementary and are left to the interested reader to verify, so let us turn to the last claim.
    Suppose, towards a contradiction, that p_k(t_0) = ṗ_k(t_0) = 0 for some 0< |t_0| < 1. In view of (<ref>) and t_0≠ 0, we obtain  

    p̈_k(t_0) = k^2 (1-t_0^2)^k-2.

    Using the recurrence relation (<ref>), we can compute from (<ref>) (and k≥ 2) that 
    
    k^2 (1-t_0^2)^k-2    = ∑_i=0^k-1(2i+2)(2i+1)a_2i+2t_0^2i
       = ∑_i=0^k-1(2k - 2i)(2k+2i+1)a_2it_0^2i + k^2 ∑_i=0^k-1 (-1)^i k-1it_0^2i,

    whence the binomial theorem yields 
    
    k^2 t_0^2 (1-t_0^2)^k-2 = ∑_i=0^k-1(2k-2i)(2k+2i+1)a_2it_0^2i = ∑_i=1^k (2k-2i)(2k+2i+1)a_2it_0^2i;

    the last identity follows from the choice a_0 = 0 and (2k-2i)(2k+2i+1)a_2i = 0 when i =k. 
    However, p_k(t_0) = ṗ_k(t_0) = 0 along with (<ref>) and t_0≠ 0 yields 
    
    ∑_i=1^k a_2it_0^2i-2 = ∑_i=1^k 2i a_2it_0^2i-2 = 0.

    By (<ref>), we can first divide by t_0^2 ≠ 0 the leftmost and rightmost sides of (<ref>) and then proceed further as 
    
    k^2 (1-t_0^2)^k-2   = ∑_i=1^k(2k-2i)(2k+2i + 1)a_2i t_0^2i - 2
            = - ∑_i=1^k 2i(2i + 1) a_2it_0^2i-2
    
                = - ∑_i=1^k 2i(2i - 1) a_2it_0^2i-2 = - k^2 t_0^2(1-t_0^2)^k-2,

where the derivation of the last identity follows simply from the first line of (<ref>). However, (<ref>) implies k^2(1+t_0^2)(1-t_0^2)^k-2 = 0, which is incompatible with |t_0| < 1. This finishes the proof. 



Finally, the next lemma gives a description of the possible free boundaries in the blow-up limit.




Let ϱ̃ be any blow-up limit as in Theorem <ref>(<ref>). Then its free boundary ∂{ϱ̃>0} is contained either in the z-axis, i.e. {r = 0}, or inside {0}∪ (Z_2k-1∖{z =0}). 



The simplest case is when the harmonic function ∂_zϱ̃ vanishes identically in ^3. Then it is easy to compute that the only such non-trivial, 2k-homogeneous, axially symmetric solution to (<ref>) is 

    ϱ̃(r,z) = 1/4k^2 r^2k.
 This is precisely the case where {ϱ̃= 0} = {r = 0}. Hence, in what follows, we assume that ∂_zϱ̃≢0 in ^3. 


As we discussed above, by differentiating both sides of (<ref>), and noting that ϱ̃= |Dϱ̃| = 0 on {ϱ̃= 0} and that ϱ̃ is a non-trivial 2k-homogeneous axially symmetric function on ^3, we observe that ∂_zϱ̃ is a zonal harmonic of degree 2k-1 on each connected component of {ϱ̃> 0} that vanishes continuously on ∂{ϱ̃> 0}. Since ∂_zϱ̃≢0, the above observation along with |Dϱ| = 0 on {ϱ = 0} shows us that {ϱ̃= 0}⊆{∂_zϱ̃= 0}⊆ Z_2k-1. Therefore, it only remains for us to show that ∂{ϱ̃> 0}∩{z = 0}∖{0} = ∅.

Suppose by contradiction that ∂{ϱ̃> 0}∩{z = 0}∖{0}≠∅. By the homogeneity of ϱ̃, we must have {z = 0}⊆∂{ϱ̃>0}. Let U be the maximal open connected set in ^3 contained in {ϱ̃> 0} and such that ∂ U contains {z = 0}. Since ∂_zϱ is a zonal harmonic of degree 2k-1 in U that vanishes on ∂ U, U cannot be the entire half-space above or below {z = 0}. Thus, by the homogeneity of ϱ̃, U is enclosed between a cone {z = t_0 √(r^2 + z^2)}, for some 0< |t_0| < 1, and the plane {z = 0}; i.e. we have (following the notation in (<ref>))

    U = _0,t_0   or    U=_t_0,0.
 
Then, it follows from our discussions at the beginning of this subsection that

    0<  ϱ̃= q   in U,     ϱ̃= |Dϱ̃| = 0   on ∂ U,

where q satisfies (<ref>)–(<ref>) with either (t_1,t_2) = (0,t_0) or (t_1,t_2) = (t_0,0). We note that, as 0<|t_0|<1, the Legendre function Q_2k of the second kind is regular in [-|t_0|, |t_0|]. By our counter assumption and the 2k-homogeneity of ϱ̃, we must have ϱ̃= |Dϱ̃| = 0 at both (r,z) = (1,0) and (r,z) = (√(1-t_0^2),t_0). Then, by (<ref>) and (<ref>),

    c_1 P_2k(t) + c_2 Q_2k(t) + p_k(t) = 0 
    
    c_1 Ṗ_2k(t) + c_2 Q̇_2k(t) + ṗ_k(t) = 0
      at t = 0, t_0  .

By (<ref>) and (<ref>), we have p_k(0) = ṗ_k(0) = 0. On the other hand, 2k being an even number implies that P_2k(0) ≠ 0, Ṗ_2k(0) = 0, Q_2k(0) = 0, and Q̇_2k(0) ≠ 0. Using these elementary facts in (<ref>) for t = 0, we deduce c_1 = c_2 = 0. Using this information back inside (<ref>), it follows that p_k(t_0) = ṗ_k(t_0) = 0. Recalling that 0 < |t_0| < 1, we arrive at a contradiction to Lemma <ref>. This shows that ∂{ϱ̃> 0}∩{z = 0} = {0 }, as desired. 



The conclusion follows immediately from Lemma <ref>.














§ TOPOLOGICAL DEGREE



In this appendix, we recall some standard properties of the local topological degree, referring the reader to <cit.> for further details.

Given Ω⊂^n, u∈ C^0(Ω;^n), and a point y∉u(∂Ω), one can associate to it an integer, which we denote by (u,Ω,y) and refer to as the local topological degree of u at y with respect to Ω.  This integer can be defined as follows. If u∈ C^1(Ω) and y is a regular value[Recall that y is a regular value of u if Du(x) is invertible for all points x∈ u^-1(y).] of u,  then

    (u,Ω,y):= ∑_x∈ u^-1(y)( D u(x));

note that, by Sard's theorem, this defines (u,Ω,·) a.e. in ^n. For maps u that are just continuous, or at points y which are not regular values, one defines (u,Ω,·) by approximation, in such a way that y↦(u,Ω,y) is continuous and (u_j,Ω,y)→(u,Ω,y) if u_j→ u  uniformly.



The topological degree has the following properties:

    
  *  normalization: (𝕀,Ω,y) is 1 if y∈Ω and 0 if y∉Ω;
    
  * 
    decomposition: if Ω_1∩Ω_2=∅ then (u,Ω_1∪Ω_2, y) = (u,Ω_1,y) + (u,Ω_2, y);
    
  *  homotopy invariance: if h∈ C^0([0,1]×Ω;^n) and y∉h([0,1]×∂Ω) then 
    λ↦(h(λ,·),Ω,y)
 is constant in [0,1];
    
  *  existence of solutions: if (u,Ω,y)≠ 0 then y∈ u(Ω);
    
  *  constancy: (u,Ω,·) is locally constant in ^n∖ u(∂Ω).



In fact, properties (<ref>)–(<ref>) characterize the topological degree uniquely.

It is also useful to have a notion of topological degree for maps defined on manifolds without boundary. If N,Ñ⊂^m are connected oriented compact manifold without boundary, then we can define the degree of a map g∈ C^1(N;Ñ) exactly as in (<ref>), namely through

    (g) := ∑_x∈ u^-1(y)( d g_x),

where y is a regular value of g and we see dg_x T_xN→ T_y Ñ as an invertible linear map.  This definition is independent of y and it coincides with the usual algebraic-topological definition of the degree: in algebraic topology,  one says that the homomorphism induced by g between the top homology groups of N and Ñ corresponds to multiplication by (g). In this paper, we are content with the differential topology definition (<ref>), and we will only apply it in the case where N=Ñ=∂Ω is the boundary of a smooth domain in ^n.








§ THE FREQUENCY FUNCTION FOR ELLIPTIC SYSTEMS





In this appendix, we recall some standard properties of the Almgren frequency for solutions of elliptic systems and some of its consequences. In this paper we are often concerned with weak solutions f∈ W^1,2(Ω;^m) of regular elliptic systems of the type

    Δ f^i = a_ij^α D_α f^j   in Ω,

for each 1≤ i≤ m.
Indeed, according to Theorem <ref>, any solution of (<ref>) is a solution of such a system near regular points.

In Section <ref> we recalled the definition of the Almgren frequency function, but other variants of this function are also possible. Following <cit.>, we define the generalized frequency through

    N(f,x_0,r):=r∫_B_r(x_0)[|Df|^2+(f-f(x_0))·Δ f ]dx/∫_∂ B_r(x_0) |f - (f)_x_0,r|^2 .

The generalized frequency is essentially comparable to the standard frequency (see the notation in Section <ref>), since

    |N(f,x_0,r)-N(f,x_0,r)|/N(f,x_0,r)≤ c r,

where c depends only on n and a_ij^α_L^∞(Ω), cf. <cit.>. However, the generalized frequency has the following advantage:


Let f∈ W^1,2(B_2;^m) be a non-constant weak solution to (<ref>).
   There are constants c > 1 and r_0 ∈ (0,1), depending only on n and a_ij^α_L^∞(B_2), such that r↦ e^c r N(f,x_0,r) is non-decreasing for all r∈ (0,r_0) and for each x_0∈ B_1. 


Proposition <ref> is proved in the scalar case in <cit.>, i.e. when m=1; however, as pointed out in <cit.>, the same calculations go through for general systems of the type (<ref>).

As a consequence of Proposition <ref>, 
the limit N(f,x_0,0^+):=lim_r→ 0 N(f,x_0,r) exists. Moreover, by the existence and uniqueness results for tangent maps proved in <cit.>, this quantity coincides with the vanishing order d of f-f(x_0) at x_0, i.e. d is the integer characterized by
   
    lim sup_x→ x_0|f(x)-f(x_0)|/|x-x_0|^d<∞,     lim sup_x→ x_0|f(x)-f(x_0)|/|x-x_0|^d+1=∞.

   In fact, near x_0, we may write
   
    f(x)-f(x_0)= P_d(x-x_0)+R(x),

   for some non-zero d-homogeneous vector-valued harmonic polynomial P_d and a remainder term R which vanishes to higher order, and satisfies the estimates

    |D^j R(x)|≤ c|x-x_0|^d - j +,   j∈{0,1},

for every ∈ (0,1), where c here may depend on . 

In a standard way, Proposition <ref> and (<ref>) also yield the following result, cf. <cit.>:


    Let f ∈ W^1,2(B_2;^m) be a weak solution of (<ref>). There is a constant c>1, depending only on n and Λ, such that 
    
    N(f,x_0,r)≤ c N(f,0,2)   for all  r∈ (0,1),

    for every x_0∈ B_1. 


The frequency controls not only the vanishing order, but also doubling properties of f. Here we state and prove a variant of the doubling property of f, which is used in Section <ref>. 


Let f ∈ W^1,2(B_2;^m) be a weak solution of (<ref>).  For every λ > 1 there exist δ,σ∈(0,1), depending only on n, m, λ, and a_ij^α_L^∞(B_2), such that if N(f,0,2) ≤λ then 

    |{ |Df|≥δ Df_L^∞(3Q)}∩ Q| ≥ (1-σ) |Q|,

for every cube Q with 3Q⊂ B_1.



By Theorem <ref> and a compactness argument we can find a small constant δ, which may depend further on λ, such that for every cube Q with 3Q⊂  B_1, 
 
    Df _L^∞(3Q)≤1/2δ Df_L^∞(1/2Q).

 The rest of the proof is now classical. By the interior estimates <cit.>, Df^i ∈ C^1,γ(2Q), for each γ∈(0,1), with 
 
    [Df^i]_C^0,γ(2Q)≤ c_γ_3Q f^i ≤ 3c_γ ( Q) D f_L^∞(3Q),

 where c_γ depends only on n, m, a_ij^α_L^∞(B_2), and γ. On the other hand, by (<ref>), there exists a point y_0∈1/2 Q such that 
 
    |Df(y_0)| ≥ 2δDf _L^∞(3Q).

 Hence, by (<ref>) with γ = 1/2, we can find a small scale r > 0, depending only on c_1/2 and δ_λ, such that 
 
    |Df| ≥δDf _L^∞(3Q)  in  Q_s(y_0),

 where s = 1/√(n)r  Q. Since y_0 ∈1/2 Q, we can choose r small so that  Q_s(y_0)⊂ Q. Thus, the assertion of the lemma holds with σ = 1- 2^-nr^n ∈ (0,1). 
 

 As a result of Lemma <ref>, we obtain an L^-γ-estimate for the energy density of solutions to regular elliptic systems. This can be considered as a simpler version of Theorem <ref>. 


    Let f ∈ W^1,2(B_2;^m) be a weak solution of (<ref>).  For every λ > 1 there exist γ > 0 and c > 1, depending only on n, m, λ, and a_ij^α_L^∞(B_2), such that if N(f,0,2) ≤λ, then

    ∫_B_1 |Df|^-γ dx ≤ c.




    The proof is essentially the same as that of Theorem <ref>, but  simpler. In fact, as Σ(f) = ∅, the critical scale r_f as in Lemma <ref> becomes irrelevant. So, we only need to use Lemma <ref> instead of Lemma <ref>, and do not need to invoke Lemma <ref> in the argument. We shall omit the details. 






§ PROOF OF LEMMA <REF>


Here we prove Lemma <ref>, which asserts the strong W^1,2-convergence of the energy minimizers among k-axially symmetric maps. Our proof follows essentially from <cit.> (see also <cit.>), which deals with harmonic maps in the axially symmetric setting. The involvement of the radial component ϱ needs extra attention, and we shall present the full detail below. 


Throughout the proof, c will denote a positive constant independent of i, and it may differ at each occurrence. Moreover, we shall use the spherical coordinate systems in the target space, to simplify the exposition; e.g., in such a system, (<ref>) can be written as

    u(r,θ,z) = (ϱ(r,z), kθ,φ(r,z)).

Furthermore, let us write D' = (∂_r,∂_z).

Upon a 0-homogeneous rescaling, we may assume that {u_i}_i=1^∞⊂ W^1,2(Ω;^3\ B^3)  is a bounded sequence for a concentric ball Ω strictly containing B^3. Then 
there exists a map u∈ W^1,2(Ω;^3) such that 

    u_i⇀ u  weakly in W^1,2(Ω;^3),

along a subsequence. By the Rellich compactness theorem, up to non-relabelled subsequences, u_i→ u strongly in L^2(Ω;^3) and u_i→ u a.e. in Ω. The almost everywhere convergence ensures that u is k-axially symmetric and that |u|≥ 1 a.e. in Ω. 

By (<ref>) and  B^3⋐Ω, we deduce from the Fubini theorem (upon slightly changing the radius of Ω through a further 0-homogeneous rescaling) that

    ∫_ S^2(|D u_i|^2 + |D u|^2 + |u_i - u|^2/λ_i^2)  d^2 ≤ c,

for a sequence λ_i→ 0.  
Let (ϱ_i,_i) be the representation of u_i, and let (ϱ,) be that of u. By (<ref>), there is λ_i'→ 0 satisfying 

    1/2λ_i^3/4≤λ_i' ≤ 2λ_i^3/4,

such that at p_i^± := (λ_i', ±√(1 - (λ_i')^2))∈𝔻, we have 

    |D'(ϱ_i,_i)(p_i^±)|^2 + |D'(ϱ,)(p_i^±)|^2 + |(ϱ_i-ϱ,_i-)(p_i^±)|^2/λ_i^2≤ c.

On the other hand, due to Lemma <ref>, ϱ_i is weakly subharmonic in Ω and, since B^3⋐Ω and {u_i} is bounded in L^2(Ω),

    ϱ_i≤ c  a.e. in  B^3;

now since ϱ_i→ϱ a.e. in Ω, (<ref>) yields

    ϱ≤ c   a.e. in  B^3.


Let v∈ W^1,2( B^3;^3\ B^3) be an arbitrary k-axially symmetric map with v=u on S^2. As in the proof of <cit.>, let us define w_i  B^3→^3 as follows. Consider the annulus _i :=  B^3 ∖ (1-λ_i) B^3, and the polar region _i := {(r,θ,z) ∈_i: r≤λ_i'√(r^2+z^2)}. Set ϱ̃_i _i→ as the linear interpolation between ϱ_i|_ S^2 and ϱ|_ S^2, i.e., 

    ϱ̃_i(r,z) := λ_i^-1(1 - |(r,z)|)ϱ((r,z)/|(r,z)|) + λ_i^-1(|(r,z)| + λ_i - 1) ϱ_i ((r,z)/|(r,z)|),

and define _i _i→ in the same way. Clearly, ρ̃_i≥ 1 and 0≤_i≤π in _i. Next, we set w̃_i _i →^3 as the 0-homogeneous extension of (ρ̃_i,kθ,_i)|_∂_i with respect to the center (0,0,±(1-λ_i/2)). Finally, define w_i B^3→^3 by

    w_i(r,θ,z)  := 
        v((1-λ_i)^-1r,kθ,(1-λ_i)^-1z)    in  B^3∖_i, 
    
        (ϱ̃_i(r,z),kθ,_i(r,z))    in _i∖_i, 
    w̃_i (r,θ,z)    in _i,
    
        u_i(r,θ,z)    in  S^2.

We refer the reader to Figure <ref> for a depiction of this construction.



Note that w_i ∈ W^1,2( B^3;^3\ B^3). We claim that 

    ∫__i |D w_i|^2  dx ≤ c λ_i^1/4.

Once the claim is proved, then we may use the minimality of u_i over the class of k-axially symmetric maps in W^1,2( B^3;^3\ B^3) to deduce that 

    ∫_ B^3 |D u_i|^2 dx ≤∫_ B^3 |D w_i|^2 dx = (1 + o(1))∫_ B^3 |D v|^2 dx + ∫__i |D w_i|^2 dx = ∫_ B^3 |D v|^2 dx + o(1).

However, by (<ref>) and the lower semicontinuity of the energy, we also have

    ∫_ B^3|D u|^2 dx ≤lim inf_i→∞∫_ B^3|D u_i|^2 dx.

The last two inequalities show that u is energy minimizing among k-axially symmetric maps, since v was an arbitrary admissible k-axially symmetric map with v=u on S^2. In fact, taking v=u,  the last two inequalities show that

    ∫_ B^3 |D u|^2 dx=lim_i→∞∫_ B^3 |D u_i|^2 dx,

and so by the weak convergence u_i ⇀ u in W^1,2( B^3;^3), we deduce that u_i → u strongly in W^1,2( B^3;^3), as desired. 

Therefore, we are only left with justifying the claim in (<ref>). Denote by _i the domain in the (r,z)-plane for which _i is obtained by the revolution of _i about the z-axis. Following the first three general (in)equalities at <cit.>, we obtain 

    |D'ϱ̃_i|^2(r,z) ≤ c [|D'ϱ_i|^2 + |D'ϱ|^2 + |ϱ_i - ϱ|^2/λ_i^2]((r,z)/|(r,z)|),

for (r,z)∈_i, whence 

    ∬__i |D'ϱ̃_i|^2 r dr dz    ≤ cλ_i ∫_∂𝔻(|D'ϱ̃_i|^2 + |D'ϱ|^2 + |ϱ_i -ϱ|^2/λ_i^2)r d^1 
       ≤ cλ_i∫_∂ B(|D u_i|^2 + |D u|^2 + |u_i - u|^2/λ_i^2) d^2 ≤ cλ_i.

Analogously, we can also compute that 

    ∬__i |D'φ̃_i|^2 r dr dz ≤ cλ_i.

On the other hand, since (r,z) ∈_i implies r > λ_i'|(r,z)|≥λ_i'(1-λ_i), it follows from (<ref>) that 

    ∬__isin^2φ̃_i/r dr dz ≤ cλ_i^-3/4 |_i| ≤ c λ_i^1/4.

However, since (<ref>) and (<ref>) yield

    __iϱ̃_i ≤ c,

combining (<ref>), (<ref>), (<ref>), and (<ref>) altogether gives   

    ∫__i∖_i |D w_i|^2 dx = ∬__i( |D'ϱ̃_i|^2 + ϱ̃_i^2 ( |D'φ̃_i|^2 + k^2/r^2sin^2φ̃_i ) ) r dr dz ≤ cλ_i^1/4.


The L^2-estimate of |D w_i| over _i is similar to the second paragraph of <cit.>. Note that the lateral part of ∂_i is the revolution of the union of the line segments γ_i^± := {t p_i^±: 1-λ_i ≤ t≤ 1}. Hence, by the definition (<ref>) of ϱ̃_i,  (<ref>) and (<ref>), we get

    |D'ϱ̃_i|^2(r,z) ≤ c[|D'ϱ_i|^2 + |D'ϱ|^2 + 1/λ_i^2|ϱ_i-ϱ|^2](p_i^±) ≤ c,

^1-a.e. for (r,z)∈γ_i^±. A similar argument for _i, using also (<ref>), gives

    |D'φ̃_i|(r,z) + k^2/r^2sin^2φ̃_i (r,z)≤ c + cλ_i^-3/2,

^1-a.e. for (r,z)∈γ_i^±. Recalling from (<ref>) the construction of w_i|_∂_i, it follows from (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) that 

    ∫__i |D w_i|^2  dx ≤ |_i| ( c + c λ_i^-3/2 ) ≤ cλ_i'λ_i ( 1 +  λ_i^-3/2 ) ≤ cλ_i^1/4.

Consequently, the claim in (<ref>) follows from (<ref>) and (<ref>), which completes the proof. 






§ UNIQUENESS OF CONTINUOUS WEAKLY CONSTRAINT MAPS


The purpose of this appendix is to establish the uniqueness of weakly constraint maps into the exterior of a ball, provided that the images lie in a small neighborhood of a point on the boundary of the ball. Our analysis follows the classical argument of Jäger–Kaul <cit.> for harmonic maps, who consider very general target manifolds without boundary. Here we give a short, self-contained proof for weakly constraint maps, which exploits the symmetry of the ball, although we believe that the same uniqueness result should hold for general target constraints. 

To be precise, let Ω be a bounded domain in ^n, let B^m⊂^m be the open unit open ball centered at the origin, and let p be an arbitrary point on S^m-1. We say that a map u∈ W^1,2(Ω; ^m\ B^m) is a weakly constraint map if it verifies 

    Δ u = - |Du|^2uχ_{|u| = 1}  in Ω

in the weak sense. For sufficiently regular maps, this system is equivalent to saying that u is a solution of the variational inequality for the Dirichlet energy <cit.>.



    Let u_i ∈ W^1,2∩ C(Ω;^m\ B^m) be weakly constraint maps, for each i∈{1,2}, such that u_i(Ω)⊂ B_r(p) for some r∈(0,1), and p ∈ S^m-1. If u_1 - u_2 ∈ W_0^1,2(Ω;^m), then u_1 = u_2 in Ω. 



Our proof essentially follows from the idea of <cit.> on harmonic maps. However, we chose to present the argument since our system (<ref>) involves a characteristic term, which may behave badly when taking the difference between two solutions. Here, the exact structure of the obstacle being a ball turns out to resolve this issue quite smoothly.

    Let us introduce 
    
    ψ:= 1/2 |u_1 - u_2|^2   and  ψ_i := 1/2 |u_i - p|^2,

    and define 
    
    w := ψ/(1-ψ_1)(1-ψ_2).

    The basic idea of the proof is that, up to a conformal change of the  metric on the domain (which depends on ψ_1,ψ_2), the function ψ is subharmonic, and so it satisfies a maximum principle.
    
    As u_i∈ C(Ω;^m\ B^m), it follows from  Theorem <ref> that u_i∈ C_^1,1(Ω;^m\ B^m). Hence, we have ψ,ψ_1,ψ_2, w∈ W^1,2∩ C_^1,1(Ω). By (<ref>), a direct computation yields 
    
    Δψ = |D(u_1-u_2)|^2 - (1 - u_1· u_2)∑_i=1^2 |Du_i|^2χ_{|u_i|=1}

    a.e. in Ω. Recall that, if f is a Sobolev function, then for every k∈ we have D^kf=0 a.e. in f^-1(0), thus
    
    Δψ = 0   a.e. on ψ^-1(0).

    On the other hand, note that |Dψ|^2 ≤ 2ψ |D(u_1-u_2)|^2 in Ω. Also, by the constraint, we have ψ≥ 1 - u_1· u_2 in Ω. Using these two inequalities in the above equation, we derive that

    Δψ≥|Dψ|^2/2ψ - ψ∑_i=1^2 |Du_i|^2χ_{|u_i|=1}  a.e. in Ω∖ψ^-1(0).

Again by the constraint we have ψ_i ≥ 1 - p· u_i in Ω, and so by (<ref>) we similarly derive that  

    Δψ_i    = |Du_i|^2 - (1- p· u_i)|Du_i|^2χ_{|u_i|=1}≥ (1 - ψ_i) |Du_i|^2 χ_{|u_i|=1}  a.e. in Ω.

One may compare (<ref>), (<ref>), and (<ref>) with <cit.>. 

As in the proof of <cit.>, let us take 

    ϕ := -∑_i=1^2log(1-ψ_i),

so that w = e^ϕψ in Ω; note from the assumption u_i (Ω)⊂ B_r(p) for some r∈(0,1) that ϕ∈ C(Ω)∩ C_^1,1(Ω) and 0≤ϕ≤ c in Ω. Define the self-adjoint elliptic operator ≡ (e^-ϕ∇· ); the ellipticity is uniform over Ω as 1≥ e^-ϕ≥ e^-c there. 

As in the proof of <cit.>, we can compute that 

    w =  (e^ϕψ) = Δψ + ψΔϕ + Dψ· Dϕ  a.e. in Ω.

As above, we have

    w = 0  a.e. in ψ^-1(0).

On the other hand, away from ψ^-1(0), we have 

    ψΔϕ + Dψ· Dϕ   =∑_i=1^2 (ψ|Dψ_i|^2/(1-ψ_i)^2 + ψΔψ_i/1-ψ_i +  Dψ· Dψ_i/1-ψ_i)
    ≥  - |Dψ|^2/2ψ + ψ∑_i=1^2 Δψ_i/1-ψ_i,

where in the last inequality we used Young's inequality

    Dψ· Dψ_i/1-ψ_i≤ψ|Dψ_i|^2/(1-ψ_i)^2 + |Dψ|^2/4ψ.

Thus, we deduce from (<ref>), (<ref>), and (<ref>) that 

    w≥Δψ - |Dψ|^2/2ψ + ψ∑_i=1^2 Δψ_i/1-ψ_i≥ 0   a.e. in Ω∖ψ^-1(0).

Combining (<ref>) with (<ref>), we conclude that 

    w ≥ 0  a.e. in Ω.

By the (uniform) ellipticity of , the weak maximum principle applies to w ∈ W^1,2∩ C_^1,1(Ω). Finally, if u_1 - u_2 ∈ W_0^1,2(Ω;^m), then w ∈ W_0^1,2(Ω), so the weak maximum principle implies w = 0 in Ω, i.e., u_1 = u_2 in Ω as desired. 


We also note here the following result, which is the analogue for constraint maps of the results of Hildebrant–Kaul–Widman for harmonic maps <cit.>:


    Let u∈ W^1,2(Ω;^m\ B^m) be a weakly constraint map.
    If u(Ω)⊂ B_r(p) for some r∈(0,1) and some p∈ S^m-1, then u is continuous.



Since 0<r<1, B_r(p)∩ S^m-1 is a graph in the direction of p. Thus we can employ <cit.> to deduce that u is continuous. Note that, although <cit.> is stated for minimizers, the short proof only relies on the Euler–Lagrange system (<ref>) and the results of <cit.>, which hold for solutions of elliptic systems.


As an immediate corollary of the above two results we obtain that a weakly constraint map which  takes values in a small neighborhood of p is the unique weakly constraint map with those boundary values: 


    Let u ∈ W^1,2(Ω;^m\ B^m) be a weakly constraint map such that u(Ω)⊂ B_r(p) for some r∈(0,1) and some p∈ S^m-1.  Then u is the unique weakly constraint map with its own boundary data, and so in particular it is minimizing.


By Theorem <ref>, u is continuous. 
    Let v∈ W^1,2_u(Ω; ^m\ B^m) be an arbitrary weakly constraint map. We claim that v∈ B_r(p) a.e. in Ω.  Indeed,  (<ref>) shows that f:= |v - p|^2 is a weakly subharmonic function in Ω. However, as v ∈ u + W_0^1,2(Ω;^m) and u(Ω)⊂ B_r(p), we have f≤ r^2 on ∂Ω, so the weak maximum principle implies that f ≤  r^2 a.e. in Ω, i.e. v∈ B_r(p) a.e. in Ω. The claim now follows from Proposition <ref>.






 

§ ACKNOWLEDGEMENTS

A.G. was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. Additionally, he is thankful to Uppsala University and KTH for their hospitality during a visit where part of this research was conducted.
S.K. was supported by a grant from the Verg Foundation.
H.S. was supported by the Swedish Research Council (grant no. 2021-03700).

 



§ DECLARATIONS


Data availability statement: All data needed are contained in the manuscript.

Funding and/or Conflicts of interests/Competing interests: The authors declare that there are no financial, competing or conflict of interests.

 

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