Rectifiability of the singular set and uniqueness of tangent cones for semicalibrated currents
Paul Minter Davide Parise Anna Skorobogatova Luca Spolaor


 
§ ABSTRACT 
We prove that the singular set of an m-dimensional integral current T in ℝ^n + m, semicalibrated by a C^2, κ_0 m-form ω is countably (m - 2)-rectifiable. Furthermore, we show that there is a unique tangent cone at ℋ^m - 2-a.e. point in the interior singular set of T. Our proof adapts techniques that were recently developed in <cit.> for area-minimizing currents to this setting. 


*
    Axel Klawonn0000-0003-4765-7387 
 Martin Lanser0000-0002-4232-9395
    
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§ INTRODUCTION
 
In this article, we study the structure of interior singularities of semicalibrated integral currents in ^m+n. Let us recall the basic definitions. 

    Let m, n ≥ 2 be positive integers. A semicalibration in ^m+n is a C^1-regular m-form such that ‖ω‖_c ≤ 1, where ‖·‖_c denotes the comass norm on Λ^m(^m+n). An m-dimensional integral current T in ^m+n (denoted T∈_m(^m+n)) is semicalibrated by ω if ω(T⃗) = 1 at ‖ T ‖-a.e. point, where T⃗ = d T/dT denotes the polar of the canonical vector measure associated to T (also denoted by T, abusing notation) and T denotes the canonical mass measure associated to T.


Note that we may assume that the ambient space is Euclidean, equipped with the Euclidean metric, in place of a sufficiently regular Riemannian manifold (as is often assumed when studying the regularity properties of area-minimizing currents). Indeed, this is because the presence of an ambient submanifold of ^m+n in which T is supported may be instead incorporated into the semicalibration; see <cit.>. 

We say that p∈(T)∖(∂ T) is an interior regular point if there exists a neighborhood of p in which T is, up to multiplicity, a smooth embedded submanifold of ^m+n. We denote the interior regular set by (T), and we refer to its complement in (T)∖(∂ T) (which is a relatively closed set) as the interior singular set, denoted by (T).

The regularity of area-minimizing currents and more specifically calibrated currents (where the semicalibrating m-form ω is closed) has been studied extensively <cit.>. Semicalibrated currents form a natural class of almost area-minimizing currents for which the underlying differential constraint has more flexibility with respect to deformations than that for calibrated currents. Typical examples of these objects are given by almost complex cycles in almost complex manifolds. Semicalibrated currents exhibit much stronger regularity properties than general almost-minimizing currents (see <cit.>), and have thus far been shown to share the same interior regularity as area-minimizing integral currents. Indeed, in the series of works <cit.> by De Lellis, Spadaro and the fourth author, it was shown that interior singularities of two dimensional semicalibrated currents are isolated, much like those of two dimensional area-minimizing integral currents. It was further shown by the fourth author in <cit.> that the interior singular set of an m-dimensional semicalibrated current has Hausdorff dimension at most m-2, which is consistent with Almgren's celebrated dimension estimate on the interior singular set of area-minimizing integral currents. In the case of special Legendrian cycles, i.e. when the ambient space is S^5 ⊂ℂ^3 and the semicalibration ω has a specific form inherited from the complex structure, it was already shown by Bellettini and Rivière that the singular set consisted only of isolated singularities <cit.>. 

The aim of this article is to further improve on these and establish a structural result for the interior singular set of semicalibrated currents, analogous to that obtained in the works <cit.>. More precisely, our main result is the following.


    Let T be an m-dimensional integral current in ^m+n, semicalibrated by a C^2,κ_0 m-form ω for some κ_0 ∈ (0,1). Then (T) is countably (m-2)-rectifiable and there is a unique tangent cone to T at ^m-2-a.e. point in (T).


This result is interesting both from a geometric and an analytic point of view. On the geometric side, calibrated submanifolds have been central objects of study in several areas of differential geometry and mathematical physics since the seminal work of Harvey and Lawson <cit.>. Two primary examples are holomorphic subvarieties and special Lagrangians in Calabi-Yau manifolds, which play an important role in string theory (especially regarding mirror symmetry, cf. <cit.>), but they also emerge naturally in gauge theory (see <cit.>). Semicalibrations are a natural generalization of calibrations, removing the condition dω = 0 on the calibrating form which is rather rigid and in particular very unstable under deformations. In fact semicalibrations were considered already in <cit.> (cf. Section 6 therein) and around the same time they became rather popular in string theory when several authors directed their attention to non-Calabi-Yau manifolds (the subject is nowadays known as “flux compactification”, cf. <cit.>): in that context the natural notion to consider is indeed a special class of semicalibrating forms (see for instance the works <cit.>, where these are called quasi calibrations). The fine structure of the singular set in the 2-dimensional case has found applications to the Castelnuovo's bound and the Gopakumar–Vafa finiteness conjecture in the recent works <cit.>.

From an analytic point of view, it exhibits a striking difference with the setting of area-minimizing currents regarding notions of frequency function. In the work <cit.>, Krummel and Wickramasekera introduced an intrinsic version of Almgren's frequency function for an area-minimizing current, known as planar frequency. Under suitable decay hypotheses, they were able to show that the planar frequency in the area-minimizing setting is almost monotone, which then played a pivotal role in their analysis of interior singularities. However, in the semicalibrated setting one does not expect almost monotonicity of the planar frequency function under the same hypotheses as in <cit.>, and indeed in Part <ref> we provide a simple counterexample demonstrating this. Intuitively, the reason for this is that the semicalibration condition is more flexible. In particular, the graph of any C^1,α single-valued function is a semicalibrated current (although with a semicalibrating form less regular than the one in Definition <ref>) and, at such a level of generality, these currents are not expected to exhibit unique continuation properties, and consequently an almost monotone planar frequency function. We have been unable to adapt the approach of Krummel and Wickramasekera to the present setting, which is ultimately why we follow the ideas in <cit.>. Whether or not one can prove Theorem <ref> utilizing the ideas in <cit.>, and in particular finding a suitable semicalibrated `planar frequency', is an interesting question.

Finally we remark that Theorem <ref> is optimal in light of recent work of Liu <cit.>.



 §.§ Structure of the article and comparison to <cit.>

In Part <ref>, we recall the singularity degree as introduced in <cit.>, and verify that its properties remain valid in the semicalibrated setting. Part <ref> is then dedicated to treating flat singular points of singularity degree strictly larger than 1, for which we may exploit the rectifiable Reifenberg methods of Naber & Valtorta, similarly to <cit.>. In Part <ref>, we then treat points of singularity degree 1 and the lower strata (the latter just for the uniqueness of tangent cones), following <cit.>. Finally, in Part <ref> we present the example that demonstrates the failure of almost-monotonicity for the intrinsic planar frequency as introduced in <cit.>, and draw some comparisons with the area-minimizing setting of <cit.>. 

Although throughout this article we mostly follow the methods of the works <cit.> of the first and third authors with Camillo De Lellis, there are a number of important differences:

    
  * Due to the presence of the semicalibration, the corresponding error term in the first variation of T must be taken into account for all variational estimates. In particular, this creates an additional term in Almgren's frequency function in this setting (see Section <ref>), which must be taken care of when establishing the BV estimate Theorem <ref>. The existence of such variational errors was already taken into consideration in the works <cit.>.
    
  * When taking coarse blow-ups (see Section <ref>), we observe that we may assume that the term dω^2_C^0 r^2-2δ_3 is infinitesimal relative to the tilt excess (T,_r) for δ_3∈ (0,δ_2); note that this subquadratic scaling is stronger than having the same assumption with the natural quadratic scaling of dω^2_C^0. The latter would be the analogue of the corresponding assumption in <cit.>, but we require this stronger assumption for Part <ref> (see Case 2 in the proof of Lemma <ref>), and we verify that it indeed holds.
    
  * We modify the original construction in <cit.> of the intervals of flattening adapted to a given geometric sequence of radii in Part <ref>, instead providing one that avoids requiring the separate treatment of the case of a single center manifold and infinitely many. This modified procedure will further be useful in the forthcoming work <cit.>.
    
  * When T is merely semicalibrated, extra care needs to be taken when applying the harmonic approximation. Indeed, note that in order to apply <cit.>, we require the stronger hypothesis dω^2_C^0≤_23(T,_1) in place of ^2 ≤(T,_1)^1/2 + 2δ in the area-minimizing case (this difference was already present in <cit.>). In particular, this affects the two regimes in the case analysis within the proof of Lemma <ref>. In order to maintain the validity of Case 1 therein, we must require that dω^2_C^0≤_23(T,_1), in place of ^3 ≤(T,_1). This in turn affects the treatment of Case 2 therein; see the second bullet point above.      
    
  * In order to obtain quadratic errors in dω_C^0 in all estimates exploiting the first variation of T in Part <ref>, we must employ an analogous absorption trick to that pointed out in <cit.> for area-minimizing currents. This makes arguments in Section <ref> more delicate.





 §.§ Acknowledgments
 
The authors would like to thank Camillo De Lellis for useful discussions. This research was conducted during the period P.M. was a Clay Research Fellow. D.P. acknowledges the support of the AMS-Simons Travel Grant, furthermore part of this research was performed while D.P. was visiting the Mathematical Sciences Research Institute (MSRI), now becoming the Simons Laufer Mathematical Sciences Institute (SLMath), which is supported by the National Science Foundation (Grant No. DMS-1928930). L.S. acknowledges the support of the NSF Career Grant DMS 2044954.



§ PRELIMINARIES AND NOTATION

Let us first introduce some basic notation. C, C_0, C_1, … will denote constants which depend only on m,n, Q, unless otherwise specified. For x∈(T), the currents T_x,r will denote the rescalings (ι_x,r)_♯ T, where ι_x,r (y):= y-x/r and ♯ denotes the pushforward. We will typically denote (oriented) m-dimensional subspaces of ^m+n (often simply referred to as planes) by π, ϖ. For x∈(T), _r(x) denotes the open (m+n)-dimensional Euclidean ball of radius r centered at p in ^m+n, while for an m-dimensional plane π⊂^m+n passing through x, B_r(x,π) denotes the open m-dimensional disk _r(x)∩π. _r(x,π) denotes the (m+n)-dimensional cylinder B_r(x,π)×π^⊥ of radius r centered at x. We let 𝐩_π: ℝ^m+n→π denote the orthogonal projection onto π, while 𝐩_π^⊥ denotes the orthogonal projection onto π^⊥. The plane π is omitted if clear from the context; if the center x is omitted, then it is assumed to be the origin. ω_m denotes the m-dimensional Hausdorff measure of the m-dimensional unit disk B_1(π). The Hausdorff distance between two subsets A and B of ^m+ n will be denoted by (A,B). Θ(T,x) denotes the m-dimensional Hausdorff density of T at x∈(T). For Q∈, _Q(^n) denotes the metric space of Q-tuples of vectors in ^n, equipped with the L^2-Wasserstein distance  (see e.g. <cit.>). Given a map f = ∑_i=1^Q  f_i taking values in _Q(^n), we use the notation η∘ f to denote the ^n-valued function 1/Q∑_i=1^Q f_i.

As for area-minimizing currents, we primarily focus our attention on the flat singular points of T, namely, those at which there exists a flat tangent cone Qπ_0 for some m-dimensional (oriented) plane π_0. By localizing around a singular point and rescaling, we may without loss of generality work under the following underlying assumption throughout.
 
m≥ 3, n ≥ 2 are integers. T is an m-dimensional integral current in _7√(m) with ∂ T_7√(m) = 0. There exists a C^2,κ_0 semicalibration ω on ^m+n such that T is semicalibrated by ω in _7√(m), with

    dω_C^1,κ_0(_7√(m))≤,

where ≤ 1 is a small positive constant which will be specified later.


Recall that if T satisfies Assumption <ref>, then in particular T is Ω-minimial as in <cit.> for some Ω >0, namely

    𝐌(T) ≤𝐌(T + ∂ S) + Ω𝐌(S),

for every S ∈𝐈_m + 1(^m+n) with compact support, and in particular one can take Ω = ‖ dω‖_C^0. In addition, if T satisfies Assumption <ref> then we have the first variation identity

    δ T(X) = T(dω X),

where X ∈ C_c^∞(ℝ^m+n∖(∂ T); ^m+n), and where δ T denotes the first variation of T:

    δ T(X) = ∫_T⃗(q) X(q)   d ‖ T ‖(q)  ,

where T⃗(q) is the oriented (approximate) tangent plane to T at q. 

Recall that the tilt excess (T,_r(x,π),ϖ) relative to an m-dimensional plane ϖ in a cylinder _r(x,π) is defined by

    (T,_r(x,π),ϖ) := 1/2ω_m r^m∫__r(x,π) |T⃗ - ϖ⃗|^2 dT.

The (optimal) tilt excess in _r(x,π) is in turn defined by

    (T,_r(x,π)) := inf_m-planes ϖ(T,_r(x,π),ϖ).

The quantities (T,_r(x),ϖ) and (T,_r(x)) are defined analogously.



 §.§ Intervals of flattening and compactness procedure

As in <cit.>, we introduce a countable collection of disjoint intervals of radii (s_j,t_j] ⊂ (0,1], for j∈∪{0} and t_0 =1, referred to as intervals of flattening, such that for _3 > 0 fixed as in <cit.> we have

    (T, _6√(m)r) ≤_3^2,     (T_0,t_j, _r) ≤ C m_0,j r^2-2δ_2    ∀ r∈(s_jt_j,3],

where

    m_0,j:=max{(T,_6√(m)t_j), ^2 t_j^2-2δ_2},

and δ_2 is fixed as in <cit.>. Observe that this definition of _0,j comes from the observation that if T is dω_C^0-minimal in _7√(m), then T_0,t_j is t_jdω_C^0-minimal in _7√(m), together with (<ref>), the estimates in <cit.> and the observation that 

    ι_0,t_j(dω)_C^0(_6√(m)) = dω∘ι_0,t_j_C^0(_6√(m)) = dω_C^0(_6√(m)t_j)≤ C dω_C^0(_6√(m)) t_j^2.

We will therefore henceforth work under the following assumption, allowing us to indeed iteratively produce the above sequence of intervals.


    T and ω are as in Assumption <ref>. The origin is a flat singular point of T and Θ(T,0) = Q∈_≥ 2. The parameter  is chosen small enough to ensure that m_0,0≤_3^2.


Following the procedure in <cit.> (see also <cit.>) with this amended choice of m_0,j, we use the center manifold construction in <cit.> to produce a sequence _j of center manifolds for the rescalings T_0,t_j with corresponding normal approximations N_j:_j →_Q(T_j^⊥), whose multigraphs agree with T_0,t_j in _3∖_s_j/t_j over an appropriately large proportion of _j∩(_3∖_s_j/t_j). By a rotation of coordinates, we may assume that the m-dimensional planes π_j over which we parameterize _j are identically equal to the same fixed plane π_0 ≡^m×{0}⊂^m+n. We refer the reader to <cit.> or <cit.> for the basic properties of the intervals of flattening. Given a center manifold  and a point x∈, we will let _r(x) denote the geodesic ball _r(x)∩ of radius r in . It will always be clear from context which particular center manifold we are using for such a ball.

Given a flat singular point of T with density Q∈ (denoted by x∈_Q(T)), we will use the terminology blow-up sequence of radii around x to refer to a sequence of scales r_k↓ 0 such that T_x,r_k_6√(m) Qπ for some m-dimensional plane π. If x=0, we will simply call this a blow-up sequence of radii, with no reference to the center. Observe that for any blow-up sequence of radii r_k, for each k sufficiently large there exists a unique choice of index j(k) such that r_k ∈ (s_j(k),t_j(k)].

Under the validity of Assumption <ref>, given a blow-up sequence of radii r_k, we will henceforth adopt the notation

    
  * T_k for the rescaled currents T_0,t_j(k)_6√(m);
    
  * _k and N_k rescpectively for the center manifolds _j(k) and the normal approximations N_j(k);
    
  * _k for the map parameterizing the center manifold _k over B_3(π_0) (see <cit.>);
    
  *  for the orthogonal projection map to  (see <cit.>).

In addition, let s=s̅_kt_j(k)∈(3 r_k/t_j(k), 3r_k/t_j(k)] be the scale at which the reverse Sobolev inequality <cit.> (see also <cit.>) holds for r=r_k/t_j(k). Then let r̅_k = 2s̅_k3t_j(k)∈(r_kt_j(k), 2r_k/t_j(k)], and in turn define the corresponding additionally rescaled objects

    T̅_k = (T_k)_0,r̅_k = (ι_0,r̅_k t_j(k))_♯ T _6√(m)r̅_k^-1,     _k = ι_0,r̅_k(_k),

together with the maps _k(x) := (x,_k(r̅_k x)) parameterizing the graphs of the rescaled center manifolds and the rescaled normal approximations N̅_k:_k→^m+n defined by

    N̅_k(x) := N_k(r̅_k x)/r̅_k.

Consequently we let u_k: B_3 ≡ B_3(π_0)→_Q(^m+n) be defined by

    u_k := N̅_k∘𝐞_k/N̅_k _L^2(_3/2),

where 𝐞_k denotes the exponential map from B_3⊂π_0≅ T_r̅_k^-1Φ_k(0)_k to  _k. In light of the reverse Sobolev inequality <cit.> implies that the sequence u_k is uniformly bounded in W^1,2(B_3/2). Then, by <cit.> (see also <cit.>), up to extracting a subsequence, there exists a Dir-minimizer u∈ W^1,2(B_3/2(π_0);_Q(π_0^⊥)) such that

    
  * η∘ u =0;
    
  * u_L^2(B_3/2)=1;
    
  * u_k → u strongly in L^2∩ W^1,2_ (B_3/2).


Recall that for Dir-minimizers u: Ω⊂^m →_Q(^m) on an open domain Ω, we may consider a regularized variant of Almgren's frequency function, defined by

    I_u(x,r) := rD_u(x,r)/H_u(x,r),      r∈ (0,(x,∂Ω))

where

    H_u(x,r) = -∫|u(y)|^2/|y-x|ϕ'(|y-x|/r)  dy,      D_u(x,r) = ∫ |Du(y)|^2 ϕ(|y-x|/r)  dy,

and ϕ:[0,∞)→[0,1] is a monotone Lipschitz function that vanishes for all t sufficiently large and is identically equal to 1 for all t sufficiently small. Similarly to the classical frequency, which formally corresponds to taking ϕ=1_[0,1], r↦ I_u(x,r) is monotone non-decreasing for each x∈Ω, and takes a constant value α if and only if u is radially α-homogeneous about x (see e.g. <cit.>). In particular, the limit

    I_u(x,0) := lim_r↓ 0 I_u(x,r)

exists and in fact is independent of the choice of ϕ. We will henceforth fix the following convenient choice of ϕ:

    ϕ (t) =
    {[    1    ,; 2-2t    ,;    0    . ].

When x=0, we omit the dependency on x for I, H and D.

Now let us define the natural regularized frequency associated to the graphical approximations for a semicalibrated current T satisfying Assumption <ref>. Given a center manifold ≡_j and a corresponding normal approximation N:→_Q(^m+n), we define the regularized frequency _N of N at a given center x∈ and scale r>0 by

    _N(x,r) := rΓ_N(x,r)/_N(x,r),

where Γ_N(x,r) = _N(x,r) + _N(x,r) with

    _N(x,r)    := -∫_|N(y)|^2/d(y,x) |∇_y d(y,x)|^2ϕ'(d(y,x)/r)  d^m(y), 
    _N(x,r)    = ∫ |DN(y)|^2 ϕ(d(y,x)/r)  d^m(y), 
    _N(x,r)    = ∑_i=1^Q ∑_l=1^m (-1)^l+1∫⟨ D_ξ_l N_i(y) ∧ξ̂_l(y) ∧ N_i(y), dω∘ι_0,t_j(y)⟩ϕ(d(y,x)/r) d^m(y).

Here, ξ̂_l = ξ_1∧⋯∧ξ_l-1∧ξ_l+1∧⋯∧ξ_m for an orthonormal frame {ξ_j}_j=1^m of T and d is the geodesic distance on the center manifold. We will often write ∇ d(x,y) to denote the derivative ∇_y d(x,y). If x=0, we will omit the dependency on the center. Recall that the presence of the additional term _N in the frequency (in contrast to that for area-minimizing integral currents) is due to the term T(dω X) in the first variation δ T(X) for T; see <cit.> for more details. 

    Note in  above the presence of the scaling ι_0,t_j. This is due to the error in the first variation being for T_0,t_j, and not for T. This is consistent with the quadratic scaling we expect for the dω_0 term in our definition of _0. 




PART: 
Singularity degree of flat singular points



§ MAIN RESULTS

Following <cit.>, we define a fine blow-up u to be any Dir-minimizer obtained through the compactness procedure in Section <ref> along a blow-up sequence of radii r_k, and we let

    (T,0) := {I_u(0) : u is a fine blow-up along some sequence r_k ↓ 0}

denote the set of frequency values of T at 0. We further recall the notion of singularity degree introduced in <cit.>:


    The singularity degree of T at 0 is defined as
    
    (T,0) := inf(T,0).


Of course, one may analogously define the set of frequency values and the singularity degree at any other point x∈(T) by instead considering fine blow-ups taken around the center point x in place of 0, and thus all of the results in this part clearly hold for any x∈_Q(T) in place of 0.

Let us now state the main result of this part, which concerns the main properties of the singularity degree.


    Suppose that T satisfies Assumption <ref>. Then
    
        
  (i) (T,0) ≥ 1 and (T,0) = {(T,0)};
        
  (ii) All fine blow-ups are radially homogeneous with degree I(T,0);
        
  (iii) if s_j_0=0 for some j_0∈ then lim_r↓ 0_N_j_0(r) = (T,0);
        
  (iv) if, conversely, there are infinitely many intervals of flattening (s_k,t_k], the functions _N_j converge uniformly to the constant function (T,0) when (T,0)>1, while when (T,0)=1, lim_k →∞_j(k)(r_kt_j(k)) = (T,0) =1 for every blow-up sequence of radii r_k;
        
  (v) if (T,0)>1 then the rescalings T_0,r converge polynomially fast to a unique flat tangent cone Qπ as r↓ 0;
        
  (vi) if additionally (T,0)> 2-δ_2 then s_j_0 = 0 for some j_0∈;
        
  (vii) if (T,0) < 2-δ_2 then there are infinitely many intervals of flattening and inf_j s_j/t_j > 0.
    


In Part 2, we will then follow the arguments in <cit.> (based on the seminal work <cit.>) to prove the following.

    Let T and ω be as in Theorem <ref>. Then the set {x∈(T): (T,x)>1} is countably (m-2)-rectifiable.


Recall that by the work <cit.> of Naber & Valtorta, for each k=0,…, m, the k-th stratum ^(k)(T), defined to be the set of all points x∈(T)∖(∂ T) such that for any tangent cone S at x we have

    ({y:(τ_y)_♯ S = S}) ≤ k,

is countably k-rectifiable. Here, τ_y(p) := p+y denotes the map that translates by y.

In Part 3 we then complete the proof of Theorem <ref> by showing the following (cf. <cit.>).
 
    Let T and ω be as in Theorem <ref>. Then the set {x∈(T): (T,x)=1} is ^m-2-null. Moreover, the tangent cone is unique at ^m-2-a.e. point in ^(m-2)(T).

Combining these results then gives Theorem <ref>.

The starting point for studying the properties of Almgren's frequency function with respect to varying normal approximations is the following result, which provides uniform upper and lower frequency bounds over all the intervals of flattening around a given flat singular point.


    Suppose that T satisfies Assumption <ref>. Then there exist constants c_0 = c_0(m,n,Q,T) > 0, C=C(m,n,Q,T)>0 and α=α(m,n,Q)>0, such that
    
    _N_j(r)    ≥ c_0     ∀ r∈(s_j/t_j,3], 
    _N_j(a)    ≤ e^Cb^α_N_j(b)     ∀ r∈(s_j/t_j,3].

    Moreover,
    
    0< inf_j inf_r∈ (s_jt_j, 3]_N_j(r) ≤sup_j sup_r∈ (s_jt_j,3]_N_j(r) < +∞.


The uniform upper bound of Theorem <ref> was established in <cit.> (see also <cit.>). For the uniform lower bound, we refer the reader to <cit.>. Although this is only proven therein in the case where T is area-minimizing, one may easily observe that the proof in fact works in exactly the same way when T is merely semicalibrated, since all of the preliminary results required (e.g. <cit.> and the estimates <cit.>) hold here also. In particular, observe that including the term _N in the frequency ensures that the variational error terms for the frequency are of exactly the same form as those in the area-minimizing case.


Given Theorem <ref>, observe that the arguments in <cit.>, rewritten for semicalibrated currents (namely, replacing all results from <cit.> with their counterparts from <cit.>), yields a local upper Minkowski dimension estimate of m-2 as obtained in <cit.> for area-minimizing integral currents.


In order to derive the conclusions of Theorem <ref>, we wish to sharpen the uniform bounds of Theorem <ref> to a quantitative control on the radial variations of the frequency function, across uninterrupted strings of intervals of flattening. With this in mind, we recall the following definition of the universal frequency from <cit.>.


    Suppose that T is as in Assumption <ref> and let {(s_k,t_k]_k=j_0^J} be a sequence of intervals of flattening with coinciding endpoints (i.e. s_k = t_k+1 for k=j_0,…,J-1), with corresponding center manifolds _k and normal approximations N_k. For r∈ (s_J, t_j_0], let
    
    (r)    := _N_k(rt_k)1_(s_k,t_k](r), 
    (r)    := _N_k(rt_k)1_(s_k,t_k](r), 
    (r)    := _N_k(rt_k)1_(s_k,t_k](r), 
    (r)    := _N_k(rt_k) 1_(s_k,t_k](r).

    We refer to  as the universal frequency function, whenever it is well-defined.


We have the following frequency BV estimate on the universal frequency function, which is not only a crucial tool for the proof of Theorem <ref>, but will also be useful in its own right in establishing the rectifiability of the points with singularity degree strictly larger than 1 in Part 2.


    There exists γ_4=γ_4(m,n,Q)>0 and C=C(m,n,Q)>0 such that the following holds. Let {(s_k,t_k]_k=j_0^J} be a sequence of intervals of flattening with coinciding endpoints. Then log( +1)∈((s_J,t_j_0]) with the quantitative estimate
    
    |[dlog(+1)/dr]_-|((s_J,t_j_0]) ≤ C ∑_k=j_0^J_0,k^γ_4.

    In addition, if (a,b]⊂ (s_k,t_k] for some interval of flattening (s_k,t_k], we have
    
    |[dlog(+1)/dr]_-|((a,b]) ≤ C (b/t_k)^γ_4_0,k^γ_4.





§ COARSE BLOW-UPS

It will be convenient to consider an alternative type of blow-up to a fine blow-up, avoiding reparameterization to center manifolds; we follow the setup of <cit.>. Consider a blow-up sequence of radii r_k and the associated sequence T_0,r_k of rescaled currents. We may assume without loss of generality that T_0,r_k Qπ_0 in _4.

    Compared to the set up of <cit.>, we are taking M = 1/2, which is sufficient for our purposes here. 


Notice that for r̅_k := r_k/t_k, in light of the stopping condition <cit.> for the intervals of flattening, we have _L⊂_2r̅_k for any Whitney cube L∈^(j(k)) with L∩B̅_r̅_k(π_0)≠∅ (see <cit.>). Let ϖ_k denote a sequence of planes such that (T_0,r_k,_4r̅_k,ϖ_k) = (T_0,r_k,_4r̅_k). Observe that for k sufficiently large, the height bound <cit.> guarantees that

    (T_0,r_k, _2,ϖ_k) ≤(T_0,r_k, _4) =: E_k → 0.

In particular, ϖ_k →π_0 (locally in Hausdorff distance). By replacing T_0,r_k by its pushforward under a rotation mapping ϖ_k to a plane parallel to π_0, which is converging to the identity, we may therefore assume that ϖ_k = π_0.

We may then ensure that for all k sufficiently large we have E_k < _1, where _1>0 is the threshold of <cit.>, which in turn yields a sequence of Lipschitz approximations 

    f_k:B_1/2(π_0)→_Q(π_0^⊥)

for T_0,r_k. Define the normalizations

    f̅_k := f_k/E_k^1/2.

We will work under the additional assumption that

    dω_C^1,κ_0^2 r_k^2-2δ_3 = o(E_k) ,

for a fixed choice of parameter δ_3 ∈ (0, δ_2).

    Notice that this is a slightly stronger hypothesis that the corresponding assumption <cit.>. The reason for asking for merely almost-quadratic scaling on the left-hand side of (<ref>) will become apparently in Part <ref>; see Remark <ref>.

Note that (<ref>) need not necessarily hold in general, but we will only need to consider cases where it is indeed true.

In light of <cit.>, we may thus conclude that up to extracting a subsequence, there exists a Dir-minimizer f̅: B_1(π_0) →_Q(π_0^⊥) with f̅(0) = Q 0 such that

    f̅_k →f̅    in W^1,2_∩ L^2(B_1(π_0)).

Recalling <cit.>, we refer to such a map f̅ as a coarse blow-up of T at 0, and we say that f̅ is non-trivial if it is not identically equal to Q 0. We in turn define the average free part

    v(x) := ∑_i=1^Q f̅_i(x) - η∘f̅(x)

for f̅. As usual, one may analogously define a coarse blow-up and its average-free part at another point x∈(T) under the assumption (<ref>).

Observe that unlike for a fine blow-up, it could be that f̅≡ Qη∘f̅. Indeed, one may construct examples of such behavior from holomorphic curves in ^2 that are of the form {(w,z): w^Q = z^p} for non-integer ratios p/Q larger than 2 (see e.g. <cit.>, and <cit.>).

We have the following frequency lower bound for coarse blow-ups, which follows from a Hardt-Simon type estimate (see <cit.>).


    Let T be as in Assumption <ref>. If f̅ is a non-trivial coarse blow-up and v is its average-free part, then I_f̅(0)≥ 1 and if v is not identically zero, then I_v(0)≥ 1.

We refer the reader to <cit.> for the proof of this, with the observation that the only differences in the argument therein are

    
  * as in <cit.>, the fact that f̅ is non-trivial is equivalent to the existence of a radius ρ>0 and a constant c̅>0 such that
    
    lim inf_k→∞(T_0,r_k,_ρ,π_k)/E_k≥c̅.

    
  * in the proof of the preliminary lemma <cit.>, the estimate (9) therein is replaced with (<ref>) here, and the Lipschitz approximation and surrounding estimates are instead taken from <cit.>;
    
  * the estimate (15) therein for a semicalibrated current follows from <cit.> in place of the classical monotonicity formula for mass ratios for stationary integral varifolds (namely, there is the presence of a higher order error term, which vanishes as the inner radius is taken to zero).


In light of Theorem <ref>, the validity of the lower bound on the singularity degree in Theorem <ref> is therefore reduced to the following.


    Suppose that T is as in Assumption <ref>. Let r_k ∈ (s_j(k),t_j(k)] be a blow-up sequence of radii with
    
    lim inf_k→∞s_j(k)/r_k >0.

    Then (<ref>) holds, and the coarse blow-up f̅ along (a subsequence of) r_k is well-defined. Moreover, for the average-free part v of f̅ and a corresponding fine blow-up u along (a further subsequence of) r_k, we have
    
    v=λ u     for some λ >0.

    In particular, I_u(0) ≥ 1.
    

    
        In the case where (<ref>) fails, we might have that v is trivial while u is not, precisely because of the possibility that f̅ = Qη∘f̅ in this case; see the above discussion.
    

    Before discussing the proof of Proposition <ref>, let us point out the following result, which one obtains as a simple consequence a posteriori, after obtaining the conclusions of Theorem <ref>, in light of the classification of homogeneous harmonic functions. This corollary will be exploited in Part 3.

    
        Let T be as in Assumption <ref> and suppose that (T,0) < 2-δ_2. Then, assuming the conclusions of Theorem <ref>, any coarse blow-up f̅ at 0 is non-trivial, average-free and (T,0)-homogeneous. Moreover, for each γ > 2((T,0) - 1) we have the lower decay bound
        
    lim inf_r↓ 0(T,_r)/r^γ > 0,

    and there exists r_0= r_0(Q,m,n, T)>0 such that
    
    (T,_r) ≥(r/s)^γ(T,_s)     ∀ r < s < r_0.

    
    Observe that given the conclusions (ii) and (vii) of Theorem <ref>, the proof of Corollary <ref> is exactly the same as that of <cit.>, when combined again with the aforementioned observation that T_0,r_j is r_jdω_C^0-minimal in _7√(m), which allows one to verify that the property (<ref>) is preserved under rescalings of blow-up sequences (see <cit.>). We thus omit the details here.

    

 §.§ Proof of Proposition <ref>

    The proof of Proposition <ref> follows analogous reasoning as that of <cit.>.

    First of all, recall the following Lemma from <cit.>. Note that it does not rely on any properties of the center manifold other than its regularity, so clearly remains unchanged in this setting.

    
    There are constants κ=κ (m,n,Q)>0 and C=C (m,n,Q)>0 with the following property. Consider:
    
        
  * A Lipschitz map g: ℝ^m ⊃ B_2 →𝒜_Q (ℝ^n) with g_C^0 +  Lip  (g) ≤κ;
        
  * A C^2 function φ : B_2 →ℝ^n with φ (0) = 0 and Dφ_C^1≤κ;
        
  * The function f (x) = ∑_i φ (x) + g_i (x) and the manifold ℳ := {(x, φ(x))};
        
  * The maps N, F: ℳ∩_3/2→𝒜_Q (ℝ^m+n) given by <cit.>, satisfying F (p) = ∑_i  p+ N_i (p), N_i(p)⊥ T_p ℳ, and T_F _5/4 = _f _5/4.
    
    If we denote by g̃ the multi-valued map x↦g̃ (x) = ∑_i  (0, g_i (x))∈𝒜_Q (ℝ^m+n), then
    
    𝒢 (N (φ (x)), g̃ (x)) ≤ C D φ_C^0 (g_C^0 + D φ_C^0)     ∀ x∈ B_1  .

    

    In addition, we also have the same comparison estimates as in <cit.>:

    
        Suppose that the assumptions of Proposition <ref> are satisfied along a sequence of radii r_k. Then (<ref>) holds, and
        
            
  (i) for _k = N̅_k_L^2(_3/2) with N̅_k as in (<ref>), we have
            
    0 < lim inf_k→∞_k^2/E_k≤lim sup_k→∞_k^2/E_k < +∞;

            
  (ii) for f_k defined as in (<ref>) and the map _k on B_2(0,π_k) such that 
            
    (_k)∩_3/2(0,π_k) = ι_0,r_k/t_j(k)(_k),

            we have
            
    ∫_B_3/2|_k- η∘ f_k|^2 = o(E_k).

        
    

    Observe that the arguments in proof of Lemma <ref> remains completely unchanged from that of <cit.>, after replacing the application of the relevant preliminary results from <cit.> with their analogues in <cit.>. In particular, we emphasize the following:
    
        
  (1) Given the estimate <cit.>, which remains valid herein since the properties of the Whitney decomposition remain unchanged, we conclude that (<ref>) holds for any δ_3∈ (0,δ_2).
        
  (2) The estimates <cit.> are replaced by <cit.> respectively; namely, despite the constructions of the respective center manifolds for area-minimizing integral currents and semicalibrated currents being different, the relevant comparison and derivative estimates are still satisfied.
    

    With Lemma <ref> and Lemma <ref> at hand, the proof of Proposition is exactly the same as that in <cit.>.

    

§ IMPROVED FREQUENCY LOWER BOUND

    This section is dedicated to the proof of the lower bound (T,0) ≥ 1 in Theorem <ref>(i). This maybe be equivalently restated as follows.

    
        Suppose that T satsfies Assumption <ref>. Then I_u(0) ≥ 1 for any fine blow-up u.
    

    The proof of Theorem <ref> follows very similar reasoning to <cit.>, now that we have Theorem <ref>, which generalizes <cit.> to the semicalibrated setting. Nevertheless, we repeat the details here.

    

 §.§ Proof of Theorem <ref>

    Let r_k ∈ (s_j(k),t_j(k)] be a blow-up sequence of radii which generates a fine blow-up u. Up to extracting a subsequence, we have three cases:
    
        
  (a) there exists J ∈ such that s_J = 0 and {r_k}⊂ (0,t_J];
        
  (b) #{j(k):k∈} = ∞ and lim_k→∞s_j(k)/r_k = 0;
        
  (c) #{j(k):k∈} = ∞ and lim_k→∞s_j(k)/r_k > 0.
    
    
    Case (a): Let , N denote respectively the center manifold and normal approximation associated to the interval of flattening (0,t_J], and let us omit dependency on N_J for  and related quantities.

    In light of the almost-monotonicity (<ref>) of Theorem <ref>, the limit I_0 := lim_r↓ 0(r) exists and lies in [c_0, ∞). Furthermore, the strong W^1,2_∩ L^2-convergence of u_k to u as in Section <ref> implies that I_u(0) = I_u(r) ≡ I_0 for each r∈ (0,3/2). It therefore remains to check that I_0 ≥ 1.

    First of all, observe that the stopping criteria for the intervals of flattening (see e.g. <cit.>) guarantees that
    
    (r) ≤ Cr^m+2-2δ_2    ∀ r∈ (0,1].

    Together with <cit.>, we additionally obtain
    
    (r) ≤ Cr^m+3-2δ_2    ∀ r∈ (0,1].

    On the other hand, observe that since (r) ≥I_0/2 for every r>0 sufficiently small, the estimates <cit.> can be rewritten as
    
    |∂_rlog((r)/r^m-1) - 2(r)/r| ≤Cr^γ_3(r)/r    ∀ r∈ (0,r_0],

    for some r_0 = r_0(I_0)>0 sufficiently small. Here, γ_3 is as in <cit.>. In particular, given >0, we have
    
    2 I_0 - /r≤∂_rlog((r)/r^m-1) ≤2 I_0 + /r    ∀ r∈(0,r_1],

    for some r_1=r_1() ∈ (0, r_0]. This in turn yields
    
    lim inf_r↓ 0(r)/r^m-1+ 2I_0 +≥(r_1)/r_1^m-1+2I_0 +  > 0,

    which then further gives the consequence
    
    lim inf_r↓ 0Γ(r)/r^m-2+ 2I_0 + > 0.

    Recalling that Γ = (r) + (r) and combining with the decay (<ref>), (<ref>), we must therefore have
    
    I_0 ≥ 2-δ_2,

    which in particular implies the desired lower bound of 1, in this case.

    Case (b): Let u denote the fine blow-up generated by (a subsequence of) r_k. Notice that
    
    I_u(ρ) = lim_k→∞_N_j(k)(ρ r_k/t_j(k))     ∀ρ∈ (0,1],

    in light of the strong W^1,2∩ L^2-convergence described in Section <ref>. In particular, since the stopping conditions for the intervals of flattening guarantee that
    
    (T,_s_j(k)) = (T_0,t_j(k),_s_j(k)/t_j(k)) ≤ C_3^2 (s_j(k)/t_j(k))^2-2δ_2⟶ 0     as k →∞,

    we deduce that {s_j(k)} is a blow-up sequence of radii. Applying Proposition <ref>, we conclude that the fine blow-up ũ generated by (a subsequence of) s_j(k) satisfies I_ũ(0) ≥ 1. Combining this with (<ref>) (which additionally holds with u replaced by ũ and r_k replaced by s_j(k)), we conclude that
    
    lim inf_k→∞_N_j(k)(s_j(k)/t_j(k)) ≥ 1.

    Combining this with the almost-monotonicity (<ref>) of the frequency, we easily conclude that for δ>0 arbitrary, there exists ρ̅>0 such that
    
    lim inf_k→∞_N_j(k)(ρ r_k/t_j(k)) ≥ 1-δ    ∀ρ∈(s_j(k)/t_j(k), ρ̅),

    from which the desired conclusion follows immediately. See <cit.> for details.
    
    Case (c): In this case, the hypotheses of Proposition <ref> hold, so applying this proposition, we immediately obtain the desired conclusion.



§ FREQUENCY BV ESTIMATE

This section is dedicated to the proof of Theorem <ref>. To begin with, we state a sharper formulation of the variational identities <cit.>. Let T satisfy Assumption <ref>, and let  be a center manifold for T with associated normal approximation N. Given x∈, set
    
    _N(x,r)    = -1/r∫_ϕ'(d(x,y)/r)∑_i N_i(y)· DN_i(y)∇ d(x,y)  dy , 
    _N(x,r)    = -1/r^2∫_ϕ'(d(x,y)/r) d(x,y)/|∇ d(x,y)|^2∑_i |DN_i(y) ·∇ d(x,y)|^2  d y , 
    Σ_N(x,r)    = ∫_ϕ(d(x,y)/r)|N(y)|^2  d y .


    
        There exist γ_4 = γ_4(m,n,Q)>0 and C = C(m,n,Q)>0 such that the following holds. Suppose that T satisfies Assumption <ref>. Let (s,t] be an interval of flattening for T around 0 with associated center manifold  and normal approximation N and let _0 be as in (<ref>) for this interval. Then _N(0,·), _N(0,·) are absolutely continuous on (st,3] and for almost every r∈ (st,3] we have
        
    ∂_r _N(0,r) = - ∫_ϕ'(d(y)/r) d(y)/r^2 |DN(y)|^2   dy 
       ∂_r _N(0,r) - m-1/r_N (0,r) = O(m_0) _N (0,r) + 2 _N(0,r), 
       |Γ_N(0,r) - _N(0,r)| ≤∑_j=1^5 |_j^o| ≤ Cm_0^γ_4_N(0,r)^1+γ_4 + Cm_0Σ_N(0,r), 
                           ≤ Cm_0^γ_4_N(0,r)^1+γ_4 + C_0 r^2 _N(0,r), 
       |∂_r _N(0,r)  - (m-2) r^-1_N(0,r)- 2_N(0,r)| ≤ 2 ∑_j=1^5 |_j^i|  + C m_0_N(0,r) 
           ≤ Cr^-1m_0^γ_4_N(0,r)^1+γ_4 + Cm_0^γ_4_N(0,r)^γ_4∂_r _N(0,r) +Cm_0 _N(0,r),
       |_N(0,r)| ≤ C_0^1/4 r _N(0,r),      |∂_r_N(0,r)| ≤ C_0^γ_4 (r^-1∂_r_N(0,r) _N(0,r))^γ_4

    where _j^o, _j^i are the variational errors as in <cit.>.
    
    The estimates in Proposition <ref> follow by the same reasoning as their weaker counterparts in <cit.>, together with the following observations:
    
        
  * the error estimates in <cit.> (more precisely, see <cit.>) can be optimized so as to gain a factor of _0^γ_4 on the right-hand side, in light of the following estimates on the geodesic distance d on each center manifold :
        
	       
  * d(x,y) = |x-y| + O(m_0^1/2 |x-y|^2), 
	       
  * |∇ d(x,y)| = 1 + O(m_0^1/2d (x,y)),
	       
  * ∇^2 (d^2) = g + O(m_0 d), where g is the metric induced on ℳ by the Euclidean ambient metric.
        
        
  * the estimates <cit.> in fact immediately yield the estimates in (<ref>), in light of (<ref>).
    
    These estimates are a simple consequence of the C^3,κ-estimates for each center manifold; see e.g. <cit.>. We therefore omit the details here.

    The estimates of Proposition <ref> in turn gives rise to the following almost-monotonicity estimate for the frequency relative to a given center manifold.

    
        There exists C=C(m,n,Q)>0 such that the following holds. Suppose that T satisfies Assumption <ref>. Let (s,t], x, , N and γ_4 be as in Proposition <ref>. Then _N(0,·) is absolutely continuous on (st,3] and for almost-every r∈ (st,3] we have
        
    ∂_r log (1 +_N(0,r)) ≥ - C_0^γ_4(1 + _N(0,r)^γ_4/r + _N(0,r)^γ_4-1∂_r_N(0,r))

    

    Now observe that <cit.> can in fact be stated for a general manifold that is the graph of a sufficiently regular function as follows, with the proof remaining completely unchanged.
    
    
        There exists a dimensional constant C = C(m,n,Q) > 0 such that the following holds. Let =(φ_r) be a C^3 m-dimensional C^3 submanifold of ^m+n, where φ_r ∈ C^3(B_r(0,π);π^⊥). Let f: B_r(0,π)→_Q(π^⊥) be a Lipschitz map. Then we have
	
    | ∫__r(0,π).   .|_f(z) - ∘𝐩(z)|^2ϕ(|𝐩_π (z)|/r)   d_f(z) - ∫_B_r(0,π)(Df, Q Dφ_r)^2   dy | 
       ≤ C∫_B_r(0,π) (|Df|^4 + |Dφ_r|^4) ϕ(|y|/r)  dy  
            +C ∫__r(0,π)| (𝐩(z)) - (φ_r(𝐩_π(z))) |   d_f(z).

    

    Thus, we immediately deduce the analogue of <cit.> when T is semicalibrated.

    
        There exists a dimensional constant C = C(m,n,Q) > 0 such that the following holds. Let T satisfy Assumption <ref>. Let (s,t] be an interval of flattening for T around 0 with corresponding center manifold  and normal approximation N, let m_0 be as in (<ref>) for (s,t]. Let φ be the parameterizing map for  over B_3(π) and let f: B_1(π) →_Q(π^⊥) be a π-approximation for T_0,t in _4(0,π) according to <cit.>. For r̅=s/t, let f_L: B_8r_L(p_L,π_L) →_Q(π_L^⊥) be a π_L-approximation for T_0,t corresponding to a Whitney cube L as in <cit.>. Let π_r̅ be such that (T_0,t,_6√(m)r̅) = (T_0,t,_6√(m)r̅, π_r̅) and let B^L := B_8r_L(p_L,π_L). Let f_r̅:B_r̅(0,π_r̅) →_Q(π_r̅^⊥) be the map reparameterizing gr  (f_L) as a graph over π_r̅ and let φ_r̅, φ_L be the maps reparameterizing (φ) as graph over π_r̅, π_L respectively. Then we have
	
    | ∫_B_1(0,π).   .(Df, Q Dφ)^2ϕ(|y|) d y - ∫__1 ∩ |DN|^2 ϕ(d(y)) d y | 
       ≤ C∫_B_1(0,π) (|Df|^4 + |Dφ|^4)d y  + Cm_0^1+γ_2 + C ∫__1 ∩ (|_|^2|N|^2 + |DN|^4)
            +C ∫__1(0,π)| (𝐩(z)) - (φ(𝐩_π(z))) |   d_f(z),

	and
	
    | ∫_B_r̅(0,π_r̅).   .(Df_r̅, Q Dφ_r̅)^2ϕ(|y|/r̅)  d y - ∫__r̅∩ |DN|^2 ϕ(d(y)/r̅)d y | 
       ≤ C∫_B_r̅(0,π_r̅) (|Df_r̅|^4 + |Dφ_r̅|^4)  d y + C∫_B^L (|Df_L|^4 + |Dφ_L|^4)  d y   
           + Cm_0^1+γ_2r̅^m+2+γ_2 + C ∫_^L (|_|^2|N|^2 + |DN|^4) 
            +C ∫__r̅(0,π_r̅)| (𝐩(z)) - (φ(𝐩_π_r̅(z))) |  d_f_r̅(z),

    where _ denotes the second fundamental form of  and γ_2 is as in <cit.>.
    

    In light of the estimates of Corollary <ref>, we further wish to control the difference between an orthogonal projection to a center manifold and the image on  of an orthogonal projection to a plane over which the center manifold is parameterized.
    
    
        There exists a constant C=C(m,n,Q) > 0 such that the following holds. Suppose that T, , m_0, r̅, f, f_r̅, π, π_r̅, φ_r̅, γ_2 are as in Corollary <ref>. Then we have
	
    ∫__r̅(0,π_r̅)| (𝐩(z)) - (φ_r̅(𝐩_π_r̅(z))) |   d_f_r̅(z) ≤ Cr̅^m+1m_0^1+γ_2, 
       ∫__1(0,π)| (𝐩(z)) - (φ(𝐩_π(z))) |   d_f(z) ≤ Cm_0^1+γ_2.

    

    The proof of Lemma <ref> follows exactly as that of <cit.>, exploiting the estimates of <cit.> and <cit.> in place of their respective analogues in <cit.>.

    We further have the following comparison estimate between neighboring center manifolds.

    
        There exists a constant C=C(m,n,Q) > 0 such that the following holds. Suppose that T satisfies Assumption <ref>. Let _k-1, _k be successive center manifolds for T associated to neighboring intervals of flattening (t_k,t_k-1] and (t_k+1, t_k] around 0. Let _k-1,_k denote their respective parameterizing maps and let N_k-1,N_k denote their normal approximations. Assume that (T_0,t_k,_6√(m),π_k) = (T_0,t_k,_6√(m)) for some plane π_k and let φ̃_k-1 be the map reparametrizing (φ_k-1) as a graph over π_k.  
	   Letting φ̃_k := φ̃_k-1(t_k/t_k-1·), we have
	   
    ∫_B_1 |Dφ_k - Dφ̃_k|^2 ≤ C m_0,k^3/2.

        and
        
    ∫_B_2 |φ_k - φ̃_k|^2 ≤ C m_0,k  .

        
        
            The proof follows the same reasoning as that of <cit.>. Nevertheless, let us provide an outline here and highlight the differences. First of all, observe that by a rotation of coordinates, we may without loss of generality assume that π_k-1=π_k≡π_0 and _k-1 = _k-1.
            
            Let η∈ C_c^∞(B_2(π_0);[0,1]) be a cutoff function satisfying η≡ 1 on B_1. Via an integration by parts, we have
            
    ∫_B_1 |Dφ_k - Dφ̃_k|^2     ≤∫_B_2 |Dφ_k - Dφ̃_k|^2 η
       = -∫_B_2 (φ_k - φ̃_k)ηΔ (φ_k - φ̃_k)  - ∫_B_2∖ B_1 Dη· (φ_k - φ̃_k) D(φ_k - φ̃_k) 
       ≤ C(m_0,k^1/2 + t_k/t_k-1m_0,k-1^1/2) ∫_B_2 |φ_k - φ̃_k|.

        
        Now recall that the construction procedure for the intervals of flattening guarantees that
        
    (t_k/t_k-1)^2-2δ_2m_0,k-1≤ C _0,k.

        Thus, (<ref>) follows from (<ref>), and so it suffices to demonstrate the latter. Given a Lipschitz approximation f_k:B_3(π_0)→_Q(π_k^⊥) for T_0,t_k_4(0,π_0) as in <cit.>, which can indeed be considered for _3 sufficiently small since (T_0,t_k,_4(0,π_0)) ≤ C_0,k, observe that it suffices to show
        
    ∫_B_2 |_k - η∘ f_k|    ≤ C_0,k, 
    ∫_B_2 |_k - η∘ f_k|    ≤ C_0,k.

        In fact, notice that (<ref>) will follow from exactly the same argument as (<ref>), when combined with (<ref>). Indeed, this is due to the fact that for f̃_k:=f_k-1(t_k·t_k-1) and f_k-1 as above but for T_0,t_k-1_4(0,π_0), we have _f_k≡_f_k≡ T_0,t_k on K×π_0^⊥ for a closed set K⊂ B_2 with 
        
    |B_2∖ K| ≤ C_0,k^1+β_0,

        where β_0>0 is as in <cit.>.

        Now let us demonstrate the validity of (<ref>). By the construction of _k, B_2 is covered by a disjoint union of the contact set Γ⊂π_0 and Whitney cubes ':={L∈: L∩ B_2≠∅}. 
        
        Therefore we have
        
    ∫_B_2 |_k -η∘ f_k|≤∫_Γ∩ B_2 |_k -η∘ f_k|_(A) + ∑_L∈'∫_L∩ B_3 |_k -η∘ f_k|_(B).

        Firstly, we have
        
    |(A)| ≤ C _0,k^1+β_0,

        due to (<ref>), together with the fact that Γ⊂ K (see <cit.>) and the estimates <cit.>. 
        
        Meanwhile, for (B), we argue as follows. For each L∈', let π_L denote the optimal plane associated to L as in <cit.>, with corresponding π_L-approximation f_L, associated tilted L-interpolating function h_L as in <cit.> and (straight) L-interpolating function g_L. Note that h_L (and hence g_L) are constructed via a different smoothing procedure here, in comparison to that in <cit.> where T is area-minimizing. More precisely, here h_L is constructed by solving a suitable PDE with boundary data η∘ f_L, while in <cit.> it is constructed via convolution of η∘ f_L. Nevertheless, by <cit.>, we still have the key estimates
        
    ∫_L |_k - g_L| ≤ C_0,kℓ(L)^m+3+β_2/3,

        for β_2>0 as in <cit.>, and
        
    ∫_B_2√(m)ℓ(L)(p_L,π_L) |h_L - η∘ f_L| ≤ C _0,kℓ(L)^m+3+β_2,

        where p_L is the center of L and ℓ(L) is the side-length of L. Combining these with a reparameterization from π_0 to π_L and the tilting estimate <cit.>, then summing over L∈', the conclusion follows; see <cit.> for the details.
    



 §.§ Proof of Theorem <ref>

    With all of the preliminary estimates of this section at hand, we are now in a position to conclude the BV estimate of Theorem <ref>. Let _k:=_N_k, _k:=_N_k, _k:=_N_k, Γ_k and let _k := r^-(m-2)_k, _k := r^-(m-1)_k, _k := r^-(m-2)_k , Γ̅_k:= _k + _k denote their respective scale-invariant quantities. Let us first consider the jumps of  at the radii t_k. We have
    
    |(t_k^+) - (t_k^-)|    =|_k-1(t_kt_k-1) - _k(1)/_k(1)| + |_k-1(t_kt_k-1) - _k(1)/_k(1)| 
           + |Γ̅_k-1(t_kt_k-1)| |1/_k-1(t_kt_k-1) - 1/_k(1)|,

    where (t_k^+):= _k-1(t_kt_k-1) and (t_k^-) := _k(1).
    Now in light of (<ref>) and (<ref>), we have
    
    |_k-1(t_kt_k-1) - _k(1)/_k(1)| ≤ C_0,k^1/4|_k-1(t_kt_k-1) - _k(1)/_k(1)|,

    and
    
    |Γ̅_k-1(t_kt_k-1)| |1/_k-1(t_kt_k-1) - 1/_k(1)| ≤ C_0,k^1/4_k-1(t_kt_k-1) |1/_k-1(t_kt_k-1) - 1/_k(1)|.

    Thus, by the exact same reasoning as that for the estimates <cit.>, we obtain
    
    |(t_k^+) - (t_k^-)| ≤ C _0,k^γ_4(1+ (t_k^+)).

    When combined with the elementary identity log w ≤ w-1 for w>0, we obtain
    
    |log(1+ (t_k^+)) - log(1+ (t_k^-))| ≤|(t_k^+) - (t_k^-)| /1+ (t_k^+)≤ C_0,k^γ_4

    On the other hand, recall from Corollary <ref> that |_(s_k,t_k) is absolutely continuous and
    
    ∂_r log(1+(r)) ≥ -C/t_k_0,k^γ_4(1+ (rt_k)^-1_k(rt_k)^γ_4 + _k(rt_k)^γ_4-1∂_r (rt_k))_=:  ν_k(r)    ∀ r∈(s_k,t_k).

    Thus, we may introduce a suitable function Ω as in <cit.>, whose distributional derivative is the measure
    
    C∑_k=j_0^J _0,k^γ_4(δ_t_k + ν_k(r)1_(s_k,t_k)^1),

    so that in addition log( + 1) +Ω is monotone non-decreasing. Since
    
    |∂_r Ω|((s_J,t_j_0]) ≤ C∑_k=j_0^J _0,k^γ_4,

    and |[∂_rlog(+1)]_-|((s_J,t_j_0]) ≤ |∂_r Ω|((s_J,t_j_0]), the proof is complete.
    

    

§ PROOF OF THEOREM <REF>

    Now that we have demonstrated that Theorem <ref> holds, we are in a position to complete the proof of the remaining conclusions (ii)-(vii) of Theorem <ref>. The starting point is the following tilt excess decay result.

    
        For any I_0> 1, there exist constants C=C(I_0,m,n,Q)>0, α=α(I_0,m,n,Q)>0 such that the following holds. Let T satisfy Assumption <ref> and suppose that (T,0)≥ I_0. Then there exists r_0=r_0(I_0,m,n,Q,T) > 0 (depending also on the center point 0) such that
        
    𝐄 (T, 𝐁_r) ≤ C (r/r_0)^αmax{𝐄 (T, 𝐁_r_0), ε̅^2 r_0^2-2δ_2}    ∀ r∈ (0,r_0)   .

        Furthermore, if C is permitted to additionally depend on α, one may choose α to be any positive number smaller than min{2((T,0)-1),2-2δ_2}.
    

    The proof of this, and all its necessary preliminaries, is the same as that of <cit.>.

    Observe that a combination of the excess decay in Proposition <ref> and the universal frequency BV estimate of Theorem <ref> immediately implies the conclusions (iii), (v), (vi) of Theorem <ref>, together with (ii), (iv) in the case when (T,0)>1. Indeed, observe that Proposition <ref> guarantees that if (T,0)>1, there exists an index j_0=j_0(r_0) large enough such that t_k+1=s_k for each k≥ j_0. Combining this with Theorem <ref> and the observation that t_k+1t_k≤ 2^-5, we obtain a uniform -estimate of the form
    
    |[dlog(+1)/dr]_-|((0,t_j_0]) ≤ C ∑_k=j_0^∞_0,k^γ_4≤ C ∑_k=j_0^∞ 2^-5αγ_4(k-j_0)_0,j_0^γ_4≤ C _0,j_0^γ_4.

    for α=α((T,0),m,n,Q)>0 as in Proposition <ref>, where C=C((T,0),m,n,Q). In particular I_0:=lim_r↓ 0(r) exists, and
    
    I_u(ρ) = I_0     ∀ρ∈(0,1],

    for every fine blow-up u. Note that in particular, if s_j_0=0 for some j_0∈, Corollary <ref> alone provides the desired conclusion that I_0=lim_r↓ 0_N_j_0(r).
    
    It remains to verify the conclusions (vi), (vii) of Theorem <ref>, as well as the conclusions (ii) and (iv) when (T,0)=1. These all follow by the same reasoning as <cit.>, with the use of the Lipschitz approximation <cit.> in place of <cit.> where needed, so we do not include the details here.
    


PART: 
Rectifiability of points with singularity degree > 1



§ SUBDIVISION
 
In this part, we prove Theorem <ref>. We will work under Assumption <ref> throughout. We will be exploiting the results of the preceding part, centered around points x∈_Q(T) (namely, applying the results therein to T_x,1). 

It will be useful to produce a countable subdivision of the set _Q(T) as follows. First of all, we may write

    _Q(T) ∩_1 = ⋃_K ∈_K,

for 

    _K := {y∈_Q (T) :  (T, y)≥ 1+2^-K}∩𝐁̅_1  .

In light of Proposition <ref>, we will further decompose each piece _K based on the initial scale r_0. Let us rewrite the statement of this proposition applied to points in _K. Note that we may ensure that C=1, up to further decreasing r_0 if necessary (dependent on the exponent α).


    Let T be as in Assumption <ref>, let K∈ and let x∈_K. For μ=2^-K-1, there exists r_0=r_0(x,m,n,Q,K)>0 such that

    𝐄 (T, 𝐁_r(x)) ≤(r/s)^2μmax{𝐄 (T, 𝐁_s(x)), ε̅^2 s^2-2δ_2}     ∀ 0<r< s < r_0  .


We may thus decompose _K as follows.


Let T be as in Assumption <ref> and let _4 be a small positive constant which will be specified later. For every K∈ℕ∖{0} define μ = μ (K) := 2^-K-1. Given K, J∈ℕ, let _K,J (which implicitly also depends on _4) denote the collection of those points x∈_K for which 

    𝐄 (T, _r (x))    ≤(r/s)^2μ𝐄 (T, _s (x))
        ∀ 0<r≤ s≤6√(m)/J
    𝐄 (T, _6 √(m) J^-1)    ≤ε_4^2.




    Observe that for each K,J∈, the set _K,J is closed, in light of upper semicontinuity of the singularity degree.


Notice that by rescaling, it suffices to prove the (m-2) rectifiability of _K,J with K∈ fixed and J=1. More precisely, the remainder of this part will be dedicated to the proof of the following.


There exists _4(m,n,Q)>0 such that the following holds. Let T be as in Assumption <ref>. Then := _K,1 (which, recall, depends on _4) is (m-2)-rectifiable and has the (m-2)-dimensional local Minkowski content bound

    |_r()| ≤ Cr^n+2    ∀ r∈ (0,1],

for some C=C(m,n, Q, T,K,γ_4,_4)>0.


    The constant C in Theorem <ref> is implicitly also dependent on a uniform bound on Almgren's frequency function _x,k over all k∈, defined relative to center manifolds that will be adapted to a given geometric sequence of scales; see Corollary <ref> below.




§ ADAPTED INTERVALS OF FLATTENING, UNIVERSAL FREQUENCY, RADIAL VARIATIONS


In order to prove Theorem <ref>, we follow a strategy which is much analogous to that in <cit.>, relying on the celebrated rectifiable Reifenberg techniques of Naber & Valtorta <cit.>. We begin by decomposing the interval of scales (0,1] around each point x∈ into countably many sub-intervals whose endpoints are given by a fixed geometric sequence and construct a center manifold for each of them, hence use it to compute a corresponding frequency function. In <cit.>, since we treat separately the points for which there are finitely many intervals of flattening, these sub-intervals are be comparable in length to the intervals of flattening. Here, however, we treat the points with finitely many intervals of flattening together with those points that have infinitely many intervals of flattening. We may do this by “artificially" stopping and restarting the center manifold procedure around each point x∈_Q(T) with (T,x)> 2-δ_2, and simply setting the new center manifold to be the rescaling of the existing one, if there is no need to change the center manifold at a given endpoint of the chosen geometric sequence of scales. We will then define a corresponding universal frequency function as in Definition <ref>, but relative to this fixed geometric sequence of scales, in place of the original intervals of flattening. The frequency variation estimates and quantitative BV estimate in the preceding part, together with the excess decay of Proposition <ref>, will be key to providing us with the necessary quantitative bounds on the points in .



 §.§ Center manifolds

Let us begin by adapting the intervals of flattening from Section <ref> around each point in , to a given fixed geometric sequence of scales.

Fix a constant γ∈ (0,12], whose choice will specified later, depending only on m, n, and Q. 

Consider a point x∈ with corresponding intervals of flattening {(t_k+1,t_k]}_k≥ 0 that have associated center manifolds _x,k and normal approximations N_x,k, together with a geometric blow-up sequence of scales {γ^j}_j≥ 0. 

For j=0, let _x,0 =_x,0 and Ñ_x,0 = N_x,0. For j=1, if γ lies in the same interval of flattening to t_0=1, let 

    _x,1 := ι_0,γ(_x,0),     Ñ_x,1(x) := N_x,0(γ x)/γ.

Otherwise, let _x,1 be the center manifold associated to T_x,γ_6√(m), with corresponding normal approximation N_x,1. 

For each j≥ 2, define _x,j inductively as follows. If γ^j lies in the same flattening to γ^j-1, let

    _x,j := ι_0,γ(_x,j-1),     Ñ_x,j(x) := N_x,j-1(γ x)/γ.

Otherwise, let _x,j be the center manifold associated to T_x,γ^j_6√(m), with corresponding normal approximation N_j.

It follows from Definition <ref> that around any x∈, we may replace the procedure in Section <ref> with the intervals (γ^k+1,γ^k] in place of (s_k,t_k], and with m_0,k therein instead defined by  

    m_x,k = 𝐄 (T_x, γ^k, _6√(m)) =
    𝐄 (T, _6√(m)γ^k (x))  .

Observe that in particular, if x∈ originally has finitely many intervals of flattening with (0,t_j_0] being the final interval, it will nevertheless have infinitely many adapted intervals of flattening, but for all k sufficiently large, _x,k and Ñ_x,k are arising as rescalings of _x,j_0 and N_x,j_0 respectively.

Abusing notation, let us henceforth simply write _x,k for the center manifold _x,k, with its corresponding normal approximation N_x,k. 

We will henceforth denote by d the geodesic distance on the center manifold ℳ_x,k, which is in fact dependent on x and k. However, since this dependence is not important and it will always be clear from context which center manifold we are taking the geodesic distance on, we will omit it. Let π_x,k denote the plane used to construct the graphical parametrization φ_x,k of the center manifold _x,k and let 𝒲^x,k denote the collection of Whitney cubes associated to _x,k as in <cit.>. Note that the center manifold ℳ_x,k does not necessarily contain the origin 0 = ι_x, γ^-k (x). However we use the point p_x,k:= (0, φ_x,k (0)) ∈π_x,k×π_x,k^⊥ as a proxy for it.

Fix η∈ (0,12], to be determined later. We observe the following simple consequence of our adapted intervals of flattening and associated center manifolds.


Let γ, η>0 be two fixed constants and let c_s = 1/64√(m). Upon choosing N_0 in <cit.> sufficiently large and adjusting accordingly the constants C_e, C_h and ε_2 in <cit.> accordingly, we can ensure that for every w∈ℳ_x,k and every r∈ [ηγ, 3], any L∈𝒲^x,k with L∩ B_r (𝐩_π_x,k (w), π_x,k)≠∅ satisfies ℓ (L) ≤ c_s r. Moreover, we have the following dichotomy. Either

    
  (a) there is a positive constant c̅_s= c̅_s(K,m,n,Q,η) ∈ (0, c_s] such that B_γ (0, π_x,k) intersects a cube L∈𝒲^x,k with ℓ (L) ≥c̅_s γ, which violates the excess condition (EX) of <cit.>;
    
  (b) _x,k+1= ι_0,γ(_x,k).




   The proof follows immediately from the contruction. 


We have the following  estimate on the universal frequency function adapted to {γ^k}_k (cf. Theorem <ref>).


	There exists (m,n,Q)∈]0,] such that for any _4∈ (0,], there exists C=C(m,n,Q,γ_4, K,γ) such that the following holds for every x∈:
	
    |[dlog (1 + (x,·))/dr]_- |([0, 1]) ≤ C∑_k  m_x,k^γ_4≤ C_x,1^γ_4



Observe that in light of Proposition <ref>, the estimate in Proposition <ref> follows by the same argument as that in Theorem <ref>. Indeed, for any k such that γ^k lies in the same original interval of flattening as γ^k-1, one may estimate the jump

    |log (1 + (x,γ^k))^+ - log (1+ (x, γ^k)^-|

due to Proposition <ref>(a) (see <cit.>). Meanwhile, if instead _x,k+1= ι_0,γ(_x,k) holds, the adapted universal frequency function is absolutely continuous on (γ^k+1,γ^k-1], and we instead simply use the variation estimate of Corollary <ref>.



 §.§ Universal frequency function
 
For each center manifold ℳ_x,k, we define the frequency function for the associated normal approximation, as in Part <ref>. We define analogous quantities to those therein, namely

    _x,k(w,r)    := ∫__x,k |D N_x,k(z)|^2 ϕ(d(w,z)/r)  dz; 
    _x,k(w,r)    := - ∫__x,k|∇ d(w,z)|^2/d(w,z) |N_x,k (z)|^2ϕ'(d(w,z)/r)   dz; 
    _x,k(w,r)    := ∑_i=1^Q ∑_l=1^m (-1)^l+1∫__x,k⟨ D_ξ_l (N_x,k)_i(z) ∧ξ̂_l(z) ∧ (N_x,k)_i(z), dω(z) ⟩ϕ(d(w,z)/r) d^m(z);
    Γ_x,k(w,r)   := _x,k(w,r)+ _x,k(w,r); 
    _x,k(w,r)    := r Γ_x,k(w,r)/_x,k(w,r)  .

We further define the quantities

    _x,k(w,r)    := -1/r∫__x,kϕ'(d(w,z)/r)∑_i (N_x,k)_i(z)· D(N_x,k)_i(z)∇ d(w,z)  dz; 
    _x,k(w,r)    := -1/r^2∫__x,kϕ'(d(w,z)/r) d(w,z)/|∇ d(w,z)|^2∑_i |D( N_x,k)_i(z) ·∇ d(w,z)|^2  dz; 
    Σ_x,k(w,r)    := ∫__x,kϕ(d(w,z)/r)|N_x,k(z)|^2  dz.


We are now in a position to introduce the universal frequency function adapted to the geometric sequence {γ^k}_k, analogously to that defined in the preceding part.


For r ∈ (γ^k+1, γ^k] and x∈, define
	
    (x, r)    := _x,k(p_x,k,rγ^k), 
    (x,r)    := _x,k(p_x,k, rγ^k), 
    (x,r)    := _x,k(p_x,k, rγ^k), 
    (x,r)    := _x,k(p_x,k, rγ^k).





 §.§ Radial frequency variations

As an immediate consequence of the total variation estimate and the fact that  is closed, we infer the existence of an uniform upper bound for the frequency 𝐈 (x,r) over all x∈. We also infer the existence of the limit 𝐈 (x,0) = lim_r↓ 0𝐈 (x,r). We can then argue as in Part <ref> to show that 𝐈 (x,0) =  (x,0) ≥ 1+ 2^-K. In turn, upon choosing ε̃ sufficiently small we infer the following.


For  as in Proposition <ref> and any _4∈ ]0,], there exists C=C(m,n,Q,γ_4,K,γ,) such that the following holds:

    1+ 2^-K-1≤𝐈 (x,r) ≤ C     ∀ x∈, ∀ r ∈ ]0, 1]  .



By a simple contradiction and compactness argument, we obtain the same consequence as that in Corollary <ref> for points sufficiently close to  at the appropriate scales.


There exists ^*∈(0,] such that for any _4∈ (0,^*] and any x∈, there exists C_0=C_0(γ,η,m,n,Q,^*,K)>0, such that the following holds for every w∈ℳ_x,k and every r ∈ (ηγ, 4]:

    C_0^-1≤𝐈_x,k (w,r) ≤ C_0  .



Let us now record the following simplified variational estimates, which may be easily deduced from those in Corollary <ref>, Proposition <ref>, but for the intervals of flattening adapted to {γ^k}_k.


Let  be as in Proposition <ref>. Suppose that T, _4, x, _x,k and N_x,k are as in Corollary <ref>. Then there exist constants C dependent on K, γ, η and  but not on x, k, such that the following estimates hold for every w ∈_x,k∩_1 and any ρ, r ∈ (ηγ, 4].

    C^-1≤_x,k (w,r) ≤ C 
       C^-1r _x,k(w,r) ≤ C^-1 r Γ_x,k (w,r) ≤_x,k (w,r) ≤ C r Γ_x,k (w,r) ≤ C r _x,k(w,r) 
       Σ_x,k (w,r) ≤ C r^2 _x,k (w,r) 
       _x,k (w,r) ≤ C _x,k (w,r) 
       _x,k(w,ρ)/ρ^m-1 = _x,k(w,r)/r^m-1exp(-C∫_ρ^r _x,k(w,s) ds/s - O (m_x,k) (r-ρ)) 
       _x,k (w, r) ≤ C _x,k (w, r/4) 
       _x,k (w,r) ≤ C r^m+3 - 2δ_2
       _x,k (w,r) ≤ C r^-1_x,k (w,r) 
       |∂_r _x,k (w,r)| ≤ C r^-1_x,k (w,r) 
       |∂_r _x,k (w,r)| ≤ C _x,k (w,r)  .

In particular:

    |Γ_x,k (w,r) - _x,k (w,r)| ≤ C m_x,k^γ_4 r^γ_4_x,k (w,r) 
       |∂_r _x,k (w,r) - m-2/r_x,k (w,r) - 2 _x,k (w,r)| ≤ C m_x,k^γ_4 r^γ_4-1_x,k (w,r) 
       |∂_r_x,k(w,r) - m-1/r_x,k(w,r) - 2_x,k(w,r)| ≤ C_x,k_x,k(w,r) 
       |_x,k(w,r)| ≤ C_x,k^γ_4 r_x,k(w,r) 
       |∂_r_x,k(w,r)| ≤ C_x,k^γ_4(r^-1∂_r_x,k(w,r)_x,k(w,r))^1/2
       ∂_r _x,k (w,r) ≥ - C m_x,k^γ_4 r^γ_4-1  .

Here, γ_4 is as in Part <ref>.


Observe that (<ref>) is the consequence of Corollary <ref>, while the remaining estimates are an easy consequence of Corollary <ref>, Proposition <ref>, combined with the construction of the sequence of adapted center manifolds and associated normal approximations. We refer the reader to <cit.> for a more in-depth explanation.



§ SPATIAL FREQUENCY VARIATIONS

A key aspect of the proof of Theorem <ref> is a quantitative control on how much a given normal approximation N = N_x,k deviates from being homogeneous on average between two scales, in terms of the frequency pinching. The latter is defined in the following way.


	Let T and  be as in Theorem <ref>, let x∈ and let _x,k and N_x,k be as in Section <ref>. Consider w ∈_x,k∩_1 and a corresponding point y = x + γ^k w. Let ρ, r>0 be two radii satisfying 
 
    ηγ^k+1≤ρ≤ r < 4γ^k  .

 We define the frequency pinching W_ρ^r(x,k,y) around y between the scales ρ and r by
	
    W_ρ^r(x,k,y) :=|_x,k(w,γ^-k r ) - _x,k(w,γ^-kρ)|.



We have the following comparison to homogeneity for the normal approximations N_x,k.

 
	Assume T and  are as in Theorem <ref>. Let x∈ and k ∈. Then there exists C = C (m,n,Q,K,γ, η) such that, for any w ∈ℳ_x,k∩_1 and any radii r,ρ>0 satisfying
  
    4 ηγ^k+1≤ρ≤ r < 2γ^k  ,

 the following holds. Let y= x + γ^k w and let _ρ/4^2r(w) :=(_2r/γ^k(w)∖_ρ/4γ^k(w))∩_x,k. Then
	
    ∫__ρ/4^2r(w)∑_i |D(N_x,k)_i(z)d(w,z)∇ d(w,z)/|∇ d(w,z)| - _x,k(w,d (w,z)) (N_x,k)_i(z)|∇ d(w,z)||^2 d z/d(w,z)
           ≤ C _x,k(w,2rγ^k) (W_ρ/8^4r(x,k,y) + m_x,k^γ_4(r/γ^k)^γ_4)log(16r/ρ).




    The argument is analogous to that of <cit.>, but taking into account the additional term _x,k in the frequency. We include the details here for the purpose of clarity.

    At the risk of abusing notation, we will omit dependency on x, w and k for all quantities, for simplicity. For instance, we simply write (s) for the quantity _x,k(w,γ^-k s), and W^4r_ρ/8 for the pinching W^4r_ρ/8(x,k,y).

    Invoking the estimates of Lemma <ref>, 
    
    we obtain
    
    W^4r_ρ/4(y)    ≥∫_ρ/4^4r∂_s (s)   ds = ∫_ρ/4^4rΓ(s) + s∂_sΓ(s) - (s)Γ(s)∂_s(s)/(s)  ds 
       ≥ 2∫_ρ/4^4rs(s)-(s)/(s)  ds - C^γ_4∫_ρ/4^4rs^γ_4(s)/(s) + s^1+γ_4(s)^2/(s)^2  ds 
       ≥ 2∫_ρ/4^4rs(s)-(s)/(s)  ds - C^γ_4((4r)^γ_4 - (ρ/4)^γ_4).

    Now notice that we may write
    
    ∫_ρ/4^4r   s(s)-(s)/(s)  ds 
       = ∫_ρ/4^4r1/s(s)∫_ -ϕ'(d(w,z)/s)1/d(w,z)(∑_j |DN_j  d(w,z)∇ d(w,z)/|∇ d(w,z)||^2. 
       .     - 2(s)∑_j N_j ·(DN_j   d(w,z)∇ d(w,z)) + (s)^2|N(z)|^2|∇ d(w,z)|^2)  d z   d s 
       = ∫_ρ/4^4r1/s(s)∫ -ϕ'(d(w,z)/s)ξ (w,z,s)/d(w,z)  d z   d s,

    where
    
    ξ(w,z,s)    = ∑_j | DN_j d(w,z)∇ d(w,z)/|∇ d(w,z)| - (s) N_j(z)|∇ d(w,z)||^2  .

    Thus,
    
    W_ρ/4^4r(y) ≥ 2∫_ρ/4^4r1/s(s)∫__s/2^s(w)ξ(w,z,s)/d(w,z)  d z   d s - Cm^γ_4(r^γ_4-ρ^γ_4).

    
    Let
    
    ζ(w,z) := ∑_j | D N_j (z)d(w,z)∇ d(w,z)/|∇ d(w,z)| - (d (w,z))  N_j(z)|∇ d(w,z)| |^2.

    We then have
    
    ζ(w,z) ≤ 2ξ(w,z,s) + 2|(s) - (d(w,z))|^2| N(z)|^2 ≤ 2ξ(w,z,s) + C W_d (w,z)^s(y) | N(z)|^2.

    Let us now control W_d (w,z)^s(y) by W_ρ/8^4r(y). In light of the quantitative almost-monotonicity (<ref>) for , for any radii ηγ^k+1 < s < t ≤γ^k we have
    
    (s) ≤(t) + Cm^γ_4 (t^γ_4- s^γ_4).

    For s∈ [ρ4,4r] this therefore yields
    
    W_d (w,z)^s(y) ≤ W_ρ/8^4r(y) +Cm^γ_4 ((4r)^γ_4 -(ρ/8)^γ_4).

    Now observe that the estimate (<ref>) gives
    
    ∫__ρ/4^2r(w)∫_d(w,z)^2d(w,z)1/s^2(s)ζ(w,z)   d s   d z    ≥1/(2r)∫__ρ/4^2r(w)ζ(w,z)∫_d(w,z)^2d(w,z)1/s^2  d s   d z 
       ≥1/2(2r)∫__ρ/4^2r(w)ζ(w,z)/d(w,z)  d z.

    Combining this with the preceding estimates, we arrive at
    
    W^4r_ρ/4(y)    + log(16r/ρ)W_ρ/8^4r(y) 
       ≥ C∫_ρ/4^4r1/s(s)∫__s/2^s(w)ζ(w,z)/d(w,z)  d z   d s - Cm^γ_4r^γ_4log(16r/ρ) - Cm^γ_4r^γ_4
       ≥ C ∫__ρ/4^2r(w)∫_d(w,z)^2d(w,z)1/s^2(s)ζ(w,z)   d s   d z - Cm^γ_4r^γ_4log(16r/ρ)  - Cm^γ_4r^γ_4
       ≥C/(2r)∫__ρ/4^2r(w)ζ(w,z)/d(w,z)  d z - Cm^γ_4r^γ_4log(16r/ρ) - Cm^γ_4r^γ_4
       ≥C/(2r)∫__ρ/4^2r(w)ζ(w,z)/d(w,z)  d z - Cm^γ_4r^γ_4log(16r/ρ) - Cm^γ_4r^γ_4.

    Rearranging and again making use of (<ref>), this yields the claimed estimate.


We will further require the following spatial variation estimate for the frequency, with control in terms of frequency pinching.


	Let T be as in Theorem <ref>, let x∈ and k ∈. Let x_1,x_2 ∈_1 ∩_x,k, y_i = x + γ^k x_i and let d(x_1,x_2) ≤γ^-k r/8, where r is such that 
	
    8η γ^k+1 < r ≤γ^k  .

	Then there exists C= C (m,n,Q,γ, η) > 0 such that for any z_1,z_2 ∈ [x_1,x_2], we have
	
    |_x,k(z_1,rγ^k) - _x,k(z_2,rγ^k)| 
           ≤ C [(W_r/8^4r(x,k,y_1))^1/2 + (W_r/8^4r(x,k,y_2))^1/2 + m_x,k^γ_4/2(rγ^k)^γ_4/2]γ^kd(z_1,z_2)/r  .



To prove Lemma <ref>, we need the following spatial variational identities for _x,k, _x,k and _x,k.


    Suppose that T is as in Theorem <ref>, let x∈ and let k ∈. Let v be a continuous vector field on _x,k. For any w∈_x,k∩_1 and any ηγ≤ r ≤ 2, letting ν_w(z):= ∇ d(w,z), we have
    
    ∂_v _x,k(w,r)    = -2/r∫__x,kϕ'(d(w,z)/r) ∑_i ⟨∂_ν_w (N_x,k)_i(z), ∂_v (N_x,k)_i(z)⟩  d^m(z) 
               
            + O(m_x,k^γ_4)r^γ_4 -1_x,k(w,r),
    ∂_v _x,k(w,r)    = - 2 ∑_i ∫__x,k|∇ d(w,z)|^2/d(w,z)ϕ'(d (w,z)/r)⟨∂_v (N_x,k)_i(z), (N_x,k)_i(z) ⟩  d^m(z), 
    
            |∂_v _x,k(w,r)|    ≤ C_x,k^1/2v_C^0(_x,k(w,r)∂_r _x,k(w,r))^1/2




    The proof of (<ref>) and (<ref>) can be found in <cit.>. To see the validity of (<ref>), omitting dependency on x,k for simplicity, we simply write
    
    ∂_v (w,r) = ∑_i=1^Q ∑_l=1^m (-1)^l+1∫_⟨ D_ξ_l N_i(z) ∧ξ̂_l(z) ∧ N_i, dω(z) ⟩ϕ'(d(w,z)/r) ∇ d(w,z)/r· v(z)   d^m(z).

    Combining with an application of Cauchy-Schwarz, this in turn yields the estimate
    
    |∂_v (w,r)| ≤ Cdω_C^0(_γ^k(x))v_C^0((w,r)∂_r (w,r))^1/2.

    Recalling (<ref>), the conclusion follows immediately.



    The majority of the proof follows in the same way as that of <cit.> (cf. <cit.>), but due to the additional error terms present when T is semicalibrated, we repeat the full argument here.
    
    We will as usual omit dependency on x,k for all objects. Let x_1,x_2 be as in the statement of the lemma and let w lie in the geodesic segment [x_1,x_2]⊂. Given a continuous vector field v on  and ρ∈ (8ηγ, 1], we have
    
    ∂_v (w,ρ) = ρ(∂_v (w,ρ) + ∂_v (w,ρ))/(w,ρ) - (w,ρ)∂_v(w,ρ)/(w,ρ).

    Let μ_w be the measure on  with density
    
    dμ_w(z) = -|∇ d(w,z)|/d(w,z)ϕ'(d(w,z)/r)d^m(z).

    Now let
    
    η_w(z) := d(w,z)∇ d(w,z)/|∇ d(w,z)| = d(w,z)/|∇ d(w,z)|ν_w(z),

    and choose v to be the vector field
    
    v(z) = d(x_1,x_2)∇ d(x_1,z)/|∇ d(x_1,z)|.

    Applying Lemma <ref> with this choice of v and exploiting the estimates of Lemma <ref>, for each ρ = rγ^-k∈ (8ηγ,1] we have
    
    ∂_v (w,ρ)    = 2/(w,ρ)∫_∑_i ⟨∂_η_w N_i, ∂_v N_i ⟩  dμ_w -2(w,ρ)/(w,ρ)∫_ |∇ d(w,·)| ∑_i ⟨∂_v N_i, N_i ⟩  dμ_w
            + C^1/2ρ^2 (w,ρ)^-1/2(∂_ρ(w,ρ))^1/2 + C^γ_4ρ^γ_4.

	Now observe that v parameterizes the geodesic line segment [x_1,x_2]⊂. Thus,
	
    ∂_v N_i(z)    = d(x_1,z) ∇ d(x_1,z)/|∇ d(x_1,z)| DN_i(z) - d(x_2,z) ∇ d(x_2,z)/|∇ d(x_2,z)| DN_i(z) 
       =∂_η_x_1 N_i(z) - ∂_η_x_2 N_i(z) 
       = (∂_η_x_1 N_i(z) - (x_1, d (x_1, z))N_i(z))_:= _1,i - (∂_η_x_2 N_i(z) - (x_2, d (x_2, z))N_i(z))_:= _2,i
            + (x_1,d (x_1, z)) - (x_2,d (x_2, z))_:= _3N_i(z).

Combining this with the above calculation and once again the estimates in Lemma <ref>, we therefore obtain

    ∂_v (w,ρ)    = 2/(w,ρ)∫_∑_i ⟨∂_η_w N_i, _1,i - _2,i⟩  d μ_w - 2(w,ρ)/(w,ρ)∫_ |∇ d(w,·)| ∑_i ⟨_1,i - _2,i, N_i ⟩ dμ_w 
            + 2/(w,ρ)∫__3∑_i ⟨∂_η_w N_i, N_i ⟩ - 2(w,ρ)/(w,ρ)∫_ |∇ d(w,·)| _3∑_i |N_i|^2 dμ_w 
            + C ^1/2ρ^2(w,ρ)^-1/2(∂_ρ(w,ρ))^1/2 + Cm^γ_4ρ^γ_4
       = 2/(w,ρ)∫_∑_i ⟨∂_η_w N_i, _1,i - _2,i⟩ d μ_w - 2(w,ρ)/(w,ρ)∫_ |∇ d(w,·)| ∑_i ⟨ N_i, _1,i - _2,i⟩ dμ_w 
            + 2_3/(w,ρ)(∫_∑_i ⟨∂_η_w N_i, N_i ⟩ dμ_w - ρ(w,ρ)) 
            + Cm^γ_4ρ^γ_4(w,ρ).

where in the last inequality, we have used that ∂_ρ(ρ) ≤1r(ρ) and (<ref>). Recalling that we aim to control the spatial frequency variation in terms of frequency pinching at the endpoints x_1 and x_2, let us rewrite _3 in the following form:

    _3    ≤ |((x_1, d (x_1,z)) -(x_1,ρ))| + |((x_1,ρ) - (x_2,ρ))| + |((x_2,ρ) - (x_2, d(x_2, z)))| 
       =  W^γ^k d (x_1, z)_r(y_1) + W_γ^k d (x_2, z)^r(y_2) + |(x_1,ρ) - (x_2,ρ)|.

Combining this with the Cauchy-Schwartz inequality and the estimates in Lemma <ref>, we have

    ∂_v (w,ρ)    ≤ C[∫∑_i(|_1,i|^2 + |_2,i|^2)   d μ_w]^1/2(1/(w,ρ)[∫∑_i |∂_η_w N_i|^2 dμ_w]^1/2 +(w,ρ)/(w,ρ)^1/2) 
            + Cm^γ_4ρ^1+γ_4|(x_1,ρ) - (x_2,ρ)|/(w,ρ)((w,ρ) + |(w,ρ)|) 
            + Cm^γ_4ρ^1+γ_4W^γ^k d (x_1, z)_r (y_1) + W_γ^k d (x_2, z)^r(y_2)/(w,ρ)((w,ρ) + |(w,ρ)|) 
            + Cm^γ_4ρ^γ_4
       ≤ C[∫∑_i(|_1,i|^2 + |_2,i|^2)   d μ_w]^1/2(1/(w,ρ)[∫∑_i |∂_η_w N_i|^2 dμ_w]^1/2 +(w,ρ)/(w,ρ)^1/2) 
            + Cm^γ_4ρ^γ_4(|(x_1,ρ) - (x_2,ρ)| + W^γ^k d (x_1, z)_r (y_1) + W_γ^k d (x_2, z)^r(y_2)) 
           + Cm^γ_4ρ^γ_4

Applying Proposition <ref>, for ℓ = 1,2 we further have

    ∫∑_i|_ℓ,i|^2   d μ_w    = - ∫∑_i|_ℓ,i|^2(z) |∇ d(w,z)|/d(w,z)ϕ'(d(w,z)/ρ)   d^m(z) 
       ≤ C(x_ℓ,2ρ) (W_r/8^4r(y_ℓ)+m^γ_4ρ^γ_4).


Together with the doubling estimate (<ref>) (which applies since d(x_ℓ,w)≤ρ) and the uniform upper frequency bound (<ref>), we thus obtain the estimate

    ∂_v (w,ρ)    ≤ C[(W_r/8^4r(y_1)+m^γ_4ρ^γ_4)^1/2 + (W_r/8^4r(y_2)+m^γ_4ρ^γ_4)^1/2] + Cm^γ_4ρ^γ_4. 
       ≤ C [W_r/8^4r(y_1)^1/2 + W_r/8^4r(y_2)^1/2] + Cm^γ_4/2ρ^γ_4/2.

Integrating this inequality over the geodesic segment [z_1,z_2]⊂, the proof is complete.
    




§ QUANTITATIVE SPINE SPLITTING

Following the notation of <cit.>, for a finite set of points X={x_0,…, x_k} we let V(X) denote the affine subspace given by

    V(X) := x_0 + ({x_1-x_0,…,x_k-x_0}).

We recall the following quantitative notions of linear independence and spanning, first introduced in <cit.>.


	We say that a set X = {x_0, x_1,…,x_k}⊂_r(w) is ρ r-linearly independent if 
	
    d(x_i, V ({x_0, …, x_i-1})) ≥ρ r

	We say that a set F ⊂_r(w) ρ r-spans a k-dimensional affine subspace V if there is a ρ r-linearly independent set of points X= {x_i}_i=0^k ⊂ F such that V= V (X).


We have the following two quantitative splitting results (cf. <cit.>).


	Suppose that T is as in Theorem <ref> and let ρ,ρ̅∈ (0,1], ρ̃∈ (η,1] be given radii. There exists ^*=^*(m,n,Q,γ,K,ρ,ρ̃,ρ̅)>0 such that for _4≤^*, the following holds. Suppose that for some x∈ and r∈ [γ^k+1,γ^k], there exists a collection of points X={x_i}_i=0^m-2⊂_r(x)∩ satisfying the properties
	
		
  * X is ρ r-linearly independent;
		
  * the nearest points z_i to x_i such that γ^-k(z_i-x) ∈_x,k satisfy
		
    W^2r_ρ̃r(x,j(k),z_i) < ^*.

		Then ∩ (_r∖_ρ̅r(V(X)))=∅.
	



	Suppose that T is as in Theorem <ref>, and let ρ,ρ̅∈ (0,1], ρ̃∈ (η,1] be given radii. For any δ>0, there exists ^†=^†(m,n,Q,γ,K,ρ,ρ̃,ρ̅,δ)∈ (0, ^*] such that for _4≤^†, the following holds. Suppose that for some x∈ and r∈ [γ^k+1,γ^k], there exists a collection of points X={x_i}_i=0^m-2⊂_r(x)∩ satisfying the properties
	
		
  * X is ρ r-linearly independent;
		
  * the nearest points z_i to x_i such that γ^-k(z_i-x) ∈_x,k satisfy
		
    W^2r_ρ̃r(x,j(k),z_i) < ^†.

	
	Then for each ζ_1,ζ_2∈_r(x)∩_^† r(V(X)) and each pair of radii r_1,r_2∈ [ρ̅,1], letting w_j denote the nearest point to γ^-k(ζ_j-x) that belongs to _x,k, the following estimate holds:
    
    |_x,k(w_1,r_1) - _x,k(w_2,r_2)| ≤δ.



Given the compactness argument in Section <ref>, the proof of both Lemma <ref> and Lemma <ref> follows in exactly the same way as that of <cit.> respectively. We therefore omit the arguments here.



§ JONES' Β_2 CONTROL AND RECTIFIABILITY

In this section, we combine all of the previous estimates of the preceding sections in this part, in order to gain control on Jones' β_2 coefficients associated to the measure ^m-2, providing a quantitative L^2-flatness control on the flat density Q singularities of T with degree strictly larger than 1.

We begin by recalling the following definition.


        Given a Radon measure μ on ^m+n, we define the (m-2)-dimensional Jones' β_2 coefficient of μ as
        
    β_2,μ^m-2(x,r) := inf_affine (m-2)-planes L[r^-(m-2)∫__r(x)((y,L)/r)^2   dμ(y)]^1/2.

    
    We have the following key estimate on β_2,μ^m-2 for a measure μ supported on .

    
    There exist α_0 = α_0(m,n,Q) > 0, η̂= η̂(m) ∈ (0, 18), =(m,n,Q,K)∈ (0,^†], C(m,n,Q,K) > 0 with the following property. Suppose that _4∈ (0,], η∈ (0,η̂] and let T and  be as in Theorem <ref>. Suppose that μ is a finite non-negative Radon measure with (μ) ⊂ and let x_0 ∈. Then for all r ∈ (8ηγ^k+1 ,γ^k] we have
    
    [β_2,μ^m-2(x_0, r/8)]^2    ≤ Cr^-(m-2)∫__r/8 (x_0) W^4r_r/8(x_0, k,𝐩_x_0, k (x))  dμ (x)
            + C m_x_0,k^α_0 r^-(m-2-α_0)μ(_r/8(x_0)).



With the estimates of Proposition <ref>, Lemma <ref> and Lemma <ref> at hand, the proof of Proposition <ref> follows by exactly the same reasoning as that of <cit.> (cf. <cit.>). We thus simply refer the reader to the argument therein.



 §.§ Proof of Theorem <ref>

The proof of rectifiability and the content bound (<ref>) follows via the same procedure as that in <cit.>, crucially making use of Proposition <ref>, the quantitative splitting results of Section <ref> and the BV estimate of Proposition <ref>. Note that in order to establish the rectifiability alone, one may make use of <cit.> in place of the rectifiable Reifenberg arguments of Naber-Valtorta, but this does not allow one to obtain the Minkowski content bound (<ref>). We do not include the details here.



PART: 
Points with singularity degree 1
In this part we conclude the proof of the main result of this work, Theorem <ref>, by showing rectifiability of the remaining part of 𝔉_Q(T), as well as the ℋ^m - 2-uniqueness of tangent cones. Namely, we prove Theorem <ref>. We follow the same outline as in <cit.>; a key preliminary result is a decay theorem for the excess to (m-2)-invariant cones formed from superpositions of planes, whenever T is much closer to such a cone than any single plane, under the assumption of no density gaps for T near the spines of such cones. Before coming to the statement of this theorem, let us first recall some notation introduced in <cit.>. We begin by defining the cones of interest. As done in the other parts, we will merely point out the differences with loc. cit, and explain the changes needed. 


Let Q≥ 2 be a fixed integer. We denote by 𝒞 (Q) those subsets of ℝ^m+n which are unions of 1 ≤ N≤ Q m-dimensional planes (affine subspaces) π_1, …, π_N 
 for which π_i ∩π_j is the same (m-2)-dimensional plane V for every pair of indices (i,j) with i<j.
 
We will use the notation 𝒫 for the subset of those elements of 𝒞 (Q) which consist of a single plane; namely, with N=1. For 𝐒∈𝒞 (Q)∖𝒫, the (m-2)-dimensional plane V described in (i) above is referred to as the spine of 𝐒 and will often be denoted by V (𝐒).


Let us now recall the conical L^2 height excess between T and elements in (Q).

	Given a ball _r(q) ⊂^m+n and a cone 𝐒∈𝒞 (Q), the one-sided conical L^2 height excess of T relative to  in _r(q), denoted (T, 𝐒, _r(q)), is defined by
	
    (T, 𝐒, _r(q)) := 1/r^m+2∫__r (q)^2 (p, 𝐒)  dT(p).

At the risk of abusing notation, we further define the corresponding reverse one-sided excess as
 
    (𝐒, T, _r (q)) := 1/r^m+2∫__r (q)∩𝐒∖_ar (V (𝐒))^2 (x,  spt  (T))  dℋ^m (x)  ,

where a=a(Q,m) is a geometric constant, to be determined later (see the discussion preceding Remark <ref>). The two-sided conical L^2 height excess is then defined by

    𝔼 (T, 𝐒, _r (q)) :=
     (T, 𝐒, _r (q)) +  (𝐒, T, _r (q))  .

We finally recall the notion of planar L^2 height excess, is given by

    ^p (T, _r (q)) = min_π∈𝒫 (q) (T, π, _r (q))  .



We may now state our key excess decay theorem. This is based on the excess decay theorem <cit.>, but in the latter, there is a built-in multiplicity one assumption, ruling out branch point singularities a priori. Such a decay theorem was more recently proven in <cit.> (see Section 13 therein and also <cit.> and <cit.>) in a higher multiplicity setting for stable minimal hypersurfaces, but in codimension 1, where one has a sheeting theorem. On the other hand, our version of this theorem is both in a higher multiplicity and higher codimension setting, thus requiring new techniques as in <cit.> that overcome the lack of sheeting.

Throughout this part, we will often work with error terms involving the quantity dω_C^0, where ω is as in Assumption <ref>. Thus, for the purpose of convenience, we will henceforth use the notation

    Ω := dω_C^0(_6√(m)).



Let δ_3 = δ_2/2, for the positive parameter δ_2 fixed as in <cit.> (cf. Parts <ref> and <ref>). For every Q,m,n, and ς>0, there are positive constants ε_0 = ε_0(Q,m,n, ς) ≤1/2, r_0 = r_0(Q,m,n, ς) ≤1/2 and C = C(Q,m,n)>0 with the following property. Assume that 


  (i) T and ω are as in Assumption <ref>;

  (ii) T (_1) ≤ (Q+1/2) ω_m;

  (iii) There is 𝐒∈𝒞 (Q)∖ such that 

    𝔼 (T, 𝐒, _1) ≤ε_0^2 𝐄^p (T, _1)

and 

    _ε_0 (ξ) ∩{p: Θ (T,p)≥ Q}≠∅    ∀ξ∈ V (𝐒)∩_1/2  ;


  (iv) Ω^2-2δ_3≤ε_0^2 𝔼 (T, 𝐒', _1) for any 𝐒'∈𝒞 (Q).

Then there is a 𝐒'∈𝒞 (Q) ∖𝒫 such that 

    
  (a) 𝔼 (T, 𝐒', _r_0) ≤ς𝔼 (T, 𝐒, _1)  

    
  (b) 𝔼 (T, 𝐒', _r_0)𝐄^p (T, _r_0)≤ 2 ς𝔼 (T, 𝐒, _1)𝐄^p (T, _1) 

    
  (c) ^2 (^'∩_1,∩_1) ≤ C 𝔼 (T, 𝐒, _1)
    
  (d) ^2 (V (𝐒) ∩_1, V (𝐒')∩_1) ≤ C 𝔼(T,𝐒,_1)^p(T,_1)  .
    


With Theorem <ref> at hand, the conclusion of Theorem <ref> follows by combining it with a covering procedure analogous to the one in <cit.>; see <cit.> for the details.



 §.§ Outline of proof of Theorem <ref>

The proof of Theorem <ref> follows the same outline as that of <cit.>. We first establish an L^2-L^∞ height bound and tilt excess estimate, analogous to <cit.>. However, all instances of  in the error terms are replaced by Ω and Ω^1-δ_3 in the height bound and tilt excess estimate respectively. This will be done in Section <ref>. 
In Section <ref> we then use the height bound to verify that the graphical parameterization results of <cit.> relative to balanced cones in (Q) (see Definition <ref>) still hold true when T is semicalibrated, again with  replaced by Ω^1-δ_3 in the errors. In Section <ref> we provide the analogues of the cone balancing results of <cit.>, which are required in order to guarantee the hypotheses on the cones ∈(Q) in order to build the graphical parameterizations of the preceding section. In Section <ref> we then verify that the Simon estimates at the spine <cit.> remain valid; the key difference is again the fact that all appearances of  in the errors become Ω^1-δ_3. In Section <ref>, we conclude with a final blow-up procedure, analogous to that in <cit.>.



§ L^2-L^∞ HEIGHT BOUND

In this section we establish Allard-type tilt-excess and L^∞ estimates relative to disjoint collections of parallel planes, analogous to those in <cit.>, but for the class of semicalibrated currents. For the remainder of this section, we make the following additional assumption.

    Q, m ≥ 3, n ≥ 2 are fixed positive integers. T and ω are as in Assumption <ref>. For some oriented m-dimensional plane π_0 ≡^m×{0}⊂^m+n passing through the origin and some positive integer Q, we have 
    
    (𝐩_π_0)_♯ T_2 = Q  B_2  ,

    and T(_2) ≤ (Q+1/2)ω_m 2^m.

The main result of this section is the following (note that a scaling argument gives the corresponding estimates for arbitrary centers and scales).

 For every 1≤ r < 2, Q, and N, there is a positive constant C̅ = C̅ (Q,m,n,N,r)>0 with the following property. Suppose that T, ω and π_0 are as in Assumption <ref>, let p_1, … , p_N ∈π_0^⊥ be distinct points, and set π:= ⋃_i p_i+π_0. 
Let

    E := ∫_𝐂_2^2 (p, π)   dT (p)  .

Then

    (T,_r, π_0)≤C̅ (E + Ω^2)

and, if E≤ 1,

    (T) ∩𝐂_r ⊂{p :  (p, π)≤C̅ (E^1/2 + Ω^1 - δ_3)}  .



A consequence of Theorem <ref> is the following, which replaces <cit.>. 

Let N be a positive integer. There is a positive constant δ = δ (Q,m,n, N) with the following properties. Assume that:

    
  (i) T and ω are as in Assumption <ref>, and for some positive r ≤1/4 and q∈(T)∩_1 we have 
    
        
  ∙ ∂ T 𝐂_4r (q) = 0;
        
  ∙ (𝐩_π_0)_♯ T _4r(q)= Q  B_4r (q);
        
  ∙ T (𝐂_2r (q)) ≤ω_m (Q+1/2) (2r)^m;
    
    
  (iii) p_1, …, p_N∈^m+n are distinct points with 𝐩_π_0 (p_i)= q and ϰ:=min{|p_i-p_j|: i<j};
    
  (iv) π_1, …, π_N are oriented planes passing through the origin with 
    
    τ := max_i |π_i-π_0|≤δmin{1, r^-1ϰ}  ;

    
  (v) Upon setting π = ⋃_i (p_i+ π_i), we have 
    
    (rΩ)^2 + (2r)^-m-2∫_𝐂_2r (q)^2 (p, π) dT≤δ^2 min{1, r^-2ϰ^2}  .


Then T𝐂_r (q) = ∑_i=1^N T_i where


  (a) Each T_i is an integral current with ∂ T_i 𝐂_r (q) = 0;

  (b) (q, π) =  (q, p_i+π_i) for each q∈ (T_i);

  (c) (𝐩_π_0)_♯ T_i = Q_i  B_r (q) for some non-negative integer Q_i.


The proof of Corollary <ref> follows verbatim the one of <cit.>, replacing <cit.> with <cit.>. 

We now recall the notion of non-oriented tilt-excess, previously introduced in <cit.>. More precisely, given an m-dimensional plane π and a cylinder 𝐂 = 𝐂_r (q, π), recall that the non-oriented tilt excess is given by

    𝐄^no (T, 𝐂):= 1/2ω_m r^m∫_𝐂 |𝐩_T - 𝐩_π|^2  dT  ,

where T (x) denotes the (approximate) tangent plane to T at x. Note that here, neither plane is oriented for the projections. In particular, we have ^no(T,) ≤ C(T,). In contrast, the reverse inequality is more subtle due to possible cancellation phenomena. Nonetheless, we have the following for semicalibrated currents.  

For every 1≤ r<2 there is a constant C̅= C̅(Q,m,n, r) such that, if T, and ω are as in Assumption <ref>, then

    𝐄 (T, 𝐂_r) ≤C̅ (𝐄^no (T, 𝐂_2) + Ω^2)  .



The proof of this follows that of <cit.> (see also <cit.>), replacing the height bound <cit.> with <cit.> in the case in which the supports of the currents are not equibounded. Furthermore, instead of Almgren's strong Lipschitz approximation for area-minimizing integral currents, we invoke its variant <cit.> for Ω-minimal currents (see Definition 1.1 therein). One can then refine the approximation in the same way to deduce the desired contradiction, and conclude the proof. 

In the case N = 1, the conclusions of Theorem <ref> are given by Allard's tilt excess estimate for varifolds with bounded generalized mean curvature (see <cit.>), together with <cit.>. It suffices to verify that the generalized mean curvature of T can be controlled uniformly by Ω, which indeed is the case by the following reasoning. Recall that T satisfies the first variation identity (<ref>) for any test vector field χ∈ C_c^∞(_6√(m);^m+n). Thus, after applying the Riesz Representation Theorem, we infer 
    
    - ∫χ·H⃗_T   d‖ T ‖ = T(dωχ) = ∫⟨ dωχ, T⃗⟩  d ‖ T ‖≤‖ dω‖_C^0∫|χ|  d‖ T ‖.

    Taking the supremum in the above, and recalling the definition of L^∞ norm in terms of its dual L^1 norm (both with respect to the local Radon measure T), we deduce that the generalized mean curvature vector H⃗_T of T satisfies the estimate
    
    ‖H⃗_T ‖_L^∞(_6√(m),T)≤‖ dω‖_C^0(_6√(m)) = Ω  .

Note that the error term in the latter is indeed quadratic in Ω, unlike Theorem 1.5 therein. In addition, note that <cit.> does not require any smallness on the tilt excess.


Theorem <ref> holds when N=1, for any Q and N.


The remainder of this section is dedicated to the proof of Theorem <ref>. First of all, observe that the results of <cit.>, handling the proof of Theorem <ref> in the case where the planes in π are well-separated, remain valid when T is semicalibrated. Indeed, they are all either reliant on the following variant of <cit.>, or are proven by analogous reasoning to it.


For every 1≤r̅<2 there is a constant σ_2 = σ_2 (Q,m,n,2-r̅)>0 with the following property. Let T and ω be as in Assumption <ref>, suppose that p_1, …, p_N∈π_0^⊥ are distinct points, and let π:=⋃_i p_i+π_0. Assume that E is as in (<ref>), let H:= min{|p_i-p_j|:i ≠ j} and suppose that 

    E  ≤σ_2          H ≥ 1  .

Then 

    spt  (T) ∩𝐂_r̅⊂{q:  (q, π)≤H/4},

and, in particular, all the conclusions of Theorem <ref> hold in _r̅. 


We defer the reader to <cit.> for the proof of this, which remains completely unchanged in the setting herein, in light of the almost-monotonicity of mass ratios that holds for all almost-minimizing currents (see, for instance, <cit.>).

As in <cit.>, we then prove the estimates (<ref>) and (<ref>) separately. For the former, we need two approximate estimates on the oriented tilt-excess; <cit.> and <cit.>, but rewritten for a semicalibrated current in ^m+n, in which case the all instances of  are replaced with Ω. More precisely, the former estimate reads as follows.


For every pair of radii 1≤ r<R ≤ 2 there are constants C̅=C̅(Q,m,n,R-r)>0 and γ = γ (Q,m,n)>0 such that the following holds. Let T and ω be as in Assumption <ref> and let π, E, H be as in Lemma <ref>. Then

    𝐄 (T, 𝐂_r)≤C̅ (E+ Ω^2) + C̅(E/H^2)^γ𝐄 (T, 𝐂_R) + C̅𝐄 (T, 𝐂_R)^1+γ  .



Observe that the majority of the proof of <cit.> is in fact written for currents with bounded generalized mean curvature with H⃗_T _L^∞≤ C, where  is the second fundamental form of the ambient Riemannian manifold. Thus, armed instead with the estimate (<ref>) (which allows us to replace ^2 by Ω^2) and replacing the use of Almgren's strong excess estimate with <cit.> we are able to follow the proof verbatim, obtaining the desired estimate (<ref>). Note that, analogously to <cit.>, we may indeed apply <cit.> since we may assume that the tilt excess (T, _r_2) falls below the threshold _21 therein, for r_2=2R+r3 as in the proof of <cit.>. Indeed, the reasoning for this remains unchanged, given Lemma <ref> (in place of <cit.>). 
The estimate of Lemma <ref> in turn yields the following bootstrapped estimate, under the assumption that E is sufficiently small relative to the minimal separation of the planes in π, which is the analogue of <cit.>.


For every pair of scales 1≤ r<r_0 < 2, there are constants C̅=C̅(Q,m,n,N,r_0-r,2-r_0)>0 and σ_4=σ_4 (Q,m,n,N,r_0-r,2-r_0)>0 with the following properties.
Let T and ω be as in Assumption <ref> and let π, E, H be as in Lemma <ref>. If in addition we have

    E ≤σ_4 min{H^2, 1}  ,

then

    𝐄 (T, 𝐂_r) ≤C̅ (E + Ω^2) + C̅(E/H^2) 𝐄 (T, 𝐂_r_0)  .



Observe that the proof of Proposition <ref> remains unchanged, given Lemma <ref>, Proposition <ref> and the semicalibrated analogues of the results from <cit.> (recall the discussion above regarding the latter). 



 §.§ Tilt excess estimate

Given Proposition <ref> and the combinatorial lemmas of <cit.>, the tilt excess estimate (<ref>) of Theorem <ref> follows exactly as in <cit.>. 



 §.§ L^2-L^∞ height bound

Following <cit.>, the proof of the L^∞-bound (<ref>) is proven by induction on N, relying on the validity of the tilt excess estimate (<ref>) that we have just established.


    Let N≥ 2 and suppose that (<ref>) holds for any N' ≤ N - 1 and any Q' ≤ Q. Then it holds for N and Q.


The key to the proof of Proposition <ref> is the following adaptation of <cit.> to the semicalibrated setting.


There are constants ρ_0 = ρ_0 (m,n,Q)>0 and C = C(Q,m,n)>0 such that, for every fixed 0<ρ≤ρ_0, there are
σ_5 = σ_5 (Q,m,n,N, ρ) ∈ (0,1] and 0 < β_0 = β_0(Q,m) < 1 such that the following holds. 
Assume T, E, and π are as in Theorem <ref> with P= {p_1, …, p_N} and that

    E + Ω^2 - 2 δ_3≤σ_5.
 
Then there is another set of points P':= {q_1, …, q_N'} with N'≤ Q such that:


  (A) (q_i, P) ≤ C (E+ Ω^2-2δ_3)^1/2 for each i;

  (B) If we set π':= ⋃ (q_i+π_0), then

    ∫_𝐂_2ρ^2 (x, π')  dT (x) 
    ≤ρ^m+2β_0 (E+ Ω^2-2δ_3).




Given Lemma <ref>, Proposition <ref>, the combinatorial lemma <cit.> and <cit.>, the proof of Proposition <ref> follows exactly by exactly the same reasoning as that in <cit.>, so we omit this concluding argument here. Observe that the semicalibrated analogue of <cit.> follows by making the same minor modifications to the proof as those in Lemma <ref>; see the discussion there for more details. 

The main difference in the proof of Lemma <ref> relative to its counterpart in the area-minimizing setting is the application of <cit.>, which yields a nearby harmonic approximation for a given strong Lipschitz approximation to T as given by <cit.>. To apply the former, instead of checking the hypothesis ≤^1/4 + δ̅ (as is done in the area-minimizing case), we need to make sure that Ω≤_23^1/2, for the geometric constant _23 therein (with η_1 fixed appropriately). We thus need to amend the case analysis within the proof of Lemma <ref> accordingly, and so we provide an outline of the proof here, for the benefit of the reader.


As in <cit.>, we divide the proof in two cases, only here it will be based on the relative sizes of := (T,_1) and Ω^2. Note that now we have the validity of the tilt-excess estimate (<ref>), i.e. ≤ C(E + Ω^2). Thus, for σ_5 small enough (depending on Q,m,n), we may assume that < _21, where _21 is as in <cit.>, allowing us to obtain a map f: B_1/4(0,π_0) →_Q(π_0^⊥) satisfying


  (i) Lip  (f) ≤ C 𝐄^β≤ C(E+Ω^2)^β;

  (ii) There is a closed set K⊂ B_1/4 with ^m(K) ≤1/2^m(B_1/4) such that 𝐆_f  (K×ℝ^n)=T (K×ℝ^n) and 

    T ((B_1/4∖ K)×ℝ^n) ≤ C (𝐄+Ω^2)^1+β≤ C(E+Ω^2)^1+β  ;


where C = C(Q,m,n), β=β(Q,m,n) ∈ (0, 12m) and we have used (<ref>) to obtain the estimates in terms of E. This in turn yields

    ∫_B_1/4 |Df|^2    ≤ C ∫_K |Df|^2 + C(+Ω^2)^1+β
       ≤ C  + C(E+Ω^2)^1+β≤ C(E + Ω^2)  .


Fix ρ_0 ∈ (0,18) to be determined at the end of Case 1 below, and fix ρ∈ (0,ρ_0] arbitrarily. Fix η_1, also to be determined at the end of Case 1 (dependent on Q,m,n,ρ). Let _23 denote the parameter of <cit.>, applied in _1 with this η_1 and the map f above taken to be the E^β-approximation therein. Note that, in particular, _23 depends on η_1 and thus on ρ. We may take σ_5 even smaller such that that < _23 (now additionally depending on ρ).

Case 1: Ω^2 ≤_23(ρ).

Applying <cit.> as mentioned above, we obtain a Dir-minimizer g:B_1/4(0,π_0)→_Q(π_0^⊥) satisfying

    ∫_B_1/4(f,g)^2    ≤η_1 ω_m ; 
    ∫_B_1/4 |Dg|^2    ≤ C(E+Ω)^2  .


We then proceed as in <cit.>, letting q_i:=g_i(0), where g_i are the distinct functions in a selection for g (possibly with multiplicities) and propagating the decay coming from the α-Hölder regularity of this Dir-minimizer g to T. This choice of q_i satisfy the desired estimate (A), and we further arrive at the final estimate

    ∫__2ρ^2(x,π')  dT(x) ≤ Cσ_5^β(E+Ω^2) + Cη_1(E+Ω^2) + Cρ^m+2α(E+Ω^2)  ,

where C=C(Q,m,n). Note that α=α(Q,m) and let β_0 = α/2. Now choose ρ_0 ≤ (3C)^1/α and η_1 ≤ρ^m+2β_0/3C. Given this choice of η_1, we may now further take σ_5 ≤(ρ^m+2β_0/3C)^1/β. This yields the estimate (B), therefore completing the proof in this case. Note that we are now fixing this choice of η_1(ρ), and therefore also fixing _23(ρ), throughout this proof.

Case 2: Ω^2 > _23(ρ)

In this case, we use the estimate (<ref>) combined with the assumption, to obtain

    ∫_B_1/4| Df |^2 ≤ C + C( + Ω^2)^1 + β≤ C( + Ω^2 + 2β) ≤ C_23^-2σ_5^2δ_3Ω^2-2δ_3 .

Combining this with the Poincaré inequality for Q-valued functions, and Hölder's inequality, we infer 

    ∫_B_1/4𝒢(f, Y) ≤ C_23^-2σ_5^2δ_3Ω^2-2δ_3  ,

for some point Y = ∑_i Q_i  q_i ∈_Q, where the q_i are distinct. Setting π':= ⋃_i (q_i+π_0) and combining (<ref>) with (ii) and <cit.> (which, as previously remarked, remains valid here, since merely almost-monotonicity of mass ratios in place of monotonicity suffices) we thus have

    ∫_𝐂_2ρ^2 (x, π')  dT (x) 
    ≤ C _23^-2σ_5^2δ_3Ω^2-2δ_3 + T ((B_1/4∖ K)×ℝ^n)
    ≤ C_23^-2σ_5^2δ_3Ω^2-2δ_3 ,

where K⊂ B_1/4 is the closed set over which T over which T is graphical, as in <cit.>. In particular, for β_0 fixed as in Case 1, we may choose σ_5≤(_23^2ρ^m+2β_0/C)^1/2δ_3.

    ∫_𝐂_2ρ^2 (x, π')  dT (x) ≤ρ^m+2β_0Ω^2-2δ_3  .

This proves conclusion (B) of the lemma, in this case. Given (<ref>), the proof of (A) in this regime follows in the same way as in <cit.>. 



    Note that Lemma <ref> is the reason behind the fact that we have Ω^2-2δ_3 in our error estimates in Theorem <ref>, and thus throughout the majority of Part 3, rather than Ω^2. Indeed, notice that Case 2 in the proof above requires the tilt excess of T to be sufficiently small relative to the relevant power of Ω; this cannot be ensured if such a power is quadratic. However, at the cost of decreasing this power slightly, we obtain the desired conclusion.




§ GRAPHICAL APPROXIMATIONS

Given Theorem <ref>, the graphical approximation results of <cit.> relative to a balanced cone follow immediately, after merely replacing any application of Almgren's strong Lipschitz approximation <cit.> with its semicalibrated variant <cit.>. We provide the main conclusions here, for clarity.

Let us begin by recalling the notions of Morgan angles and M-balanced cones, which are key for the results in this section.


Given two m-dimensional linear subspaces α, β of ℝ^m+n whose intersection has dimension m-2, we consider the two positive eigenvalues λ_1 ≤λ_2 of the quadratic form Q_1: α→ given by Q_1(v) := ^2(v,β). The Morgan angles of the pair α and β are the numbers θ_i (α, β) := arcsin√(λ_i) for i=1,2.

Let M≥ 1, N∈. We say that = α_1∪⋯∪α_N∈(Q) is M-balanced if for every i≠ j, the inequality

    θ_2 (α_i, α_j) ≤ M θ_1 (α_i, α_j)

holds for the two Morgan angles of the pair α_i, α_j.


Furthermore, for = α_1∪⋯∪α_N∈(Q), recall the following notation for the minimal separation between the planes in :

    σ() := min_1 ≤ i<j ≤ N(α_i∩_1,α_j∩_1)  .

We additionally recall the layering subdivision of <cit.>. More precisely, we apply <cit.> with a parameter δ̅, to be fixed in Assumption <ref> below, in place of δ therein. This yields a family of sub-cones 𝐒 = 𝐒_0 ⊋𝐒_1 ⊋⋯⊋𝐒_κ where 𝐒_k consists of the union of the planes α_i with i∈ I (k) for the set of indices I(k) therein. We then distinguish two cases:


  (a) if max_i<j∈ I (κ) (α_i ∩_1, α_j ∩_1) < δ̅, we define an additional cone 𝐒_κ+1 consisting of a single plane, given by the smallest index in I (κ) and we set κ̅:= κ+1 and I(κ̅):= {min I(κ)};

  (b) otherwise, we select no smaller cone and set κ̅:= κ.


In this section, we work with the following underlying assumption.


Suppose T and ω are as in Assumption <ref> and T (_4) ≤ 4^m (Q+1/2) ω_m. Suppose 𝐒=α_1∪⋯∪α_N ∈𝒞 (Q)∖𝒫 is M-balanced, where M≥ 1 is a given fixed constant. 

Firstly, we denote by δ^* the minimum of the parameters δ needed to ensure that the semicalibrated versions (in ambient space ^m+n) of <cit.> are applicable to all the cones 𝐒_k, k∈{0,1,…,κ̅}; note that all the _k are M-balanced by construction and that therefore δ^* = δ^*(m,n,Q,M)>0. Subsequently, we fix a parameter τ = τ(m,n,Q,M)>0 smaller than c δ^* for the small constant c = c(m,n,Q)>0 determined by <cit.> (with n̅ = n and Σ = ^m+n); note that this constant remains completely unchanged in the setting herein. 

We then fix the parameter δ̅ smaller than c τ for this same constant c; so δ̅= δ̅(m,n,Q,M)>0. In particular, δ̅≤δ^*. Finally, ε = ε(m,n,Q,δ^*,δ̅,τ)>0 is determined in Proposition <ref> below, and will be smaller than both c δ̅ for the same parameter c above, and the parameter  of <cit.>. Note that  is implicitly additionally dependent on the two parameters σ,ς of Proposition <ref>, which are fixed arbitrarily. We assume that {Θ (T,·) ≥ Q}∩_ε (0) ≠∅ and suppose that

    𝔼 (T, 𝐒, _4) + Ω^2-2δ_3≤ε^2 σ ()^2  ,

where 𝔼(T,,_4) is defined as in Definition <ref>.




 §.§ Whitney decomposition

We recall here the main aspects of the Whitney decomposition of <cit.> and the associated notation. We refer the reader therein for more details, including figures illustrating the decomposition and associated regions in the ambient space. 

Let =α_1∪⋯∪α_N∈(S). Let L_0 be the closed cube in V=V() with side-length 2/√(m-2) centered at 0 and let R be the rotationally invariant (around V) region given by

    R :={p: 𝐩_V (p) ∈ L_0  and   0<|𝐩_V^⊥ (p)|≤ 1}  .

We recall here that we are assuming m≥ 3 (cf. Assumption <ref>); note that this is the only reason why such a restriction is necessary. 

For every ℓ∈ℕ denote by 𝒢_ℓ the collection of (m-2)-dimensional cubes in the spine V obtained by subdividing L_0 into 2^ℓ (m-2) cubes of side-length 2^1-ℓ/√(m-2), and we let 𝒢 = ⋃_ℓ𝒢_ℓ. We write L for a cube in 𝒢, so L∈𝒢_ℓ for some ℓ∈. When we want to emphasize the dependence of the integer ℓ on L we will write ℓ (L). We use the standard terminology parent, child, ancestor, descendant to describe relations of cubes; see <cit.> for details. For every L∈𝒢_ℓ we let 

    R (L) := {p: 𝐩_V (p)∈ L      2^-ℓ-1≤ |𝐩_V^⊥ (p)|≤ 2^-ℓ}  .

For each L∈𝒢_ℓ we let y_L∈ V be its center and denote by (L) the ball 𝐁_2^2-ℓ(L) (y_L) (in ^m+n)
and by 𝐁^h (L) the set (L) ∖ B_ρ_*2^-ℓ(L) (V), where ρ_* is as in <cit.>. We identify three mutually dijoint subfamilies of cubes in ; outer cubes, central cubes and inner cubes, defined precisely in <cit.>. These families of cubes will be denoted by 𝒢^o, 𝒢^c, and 𝒢^in, respectively. By construction, any cube L∈𝒢 is either an outer cube, or a central cube, or an inner cube, or a descendant of inner cube. We in turn define three subregions of R:

    
  * The outer region, denoted R^o, is the union of R (L) for L varying over elements of 𝒢^o.
    
  * The central region, denoted R^c, is the union of R (L) for L varying over elements of 𝒢^c.
    
  * Finally, the inner region, denoted R^in, is the union of R(L) for L ranging over the elements of 𝒢 which are neither outer nor central cubes.


For every i∈{1, …, N} we further define 

    R^o_i := ⋃_L∈𝒢^o L_i ≡α_i∩⋃_L∈𝒢^o R (L)

and let Q_i := Q_L_0, i. 

We refer the reader to <cit.> for key properties about the Whitney cubes and the conical excess of T associated to each of them; the conclusions remain unchanged herein. 

Observe that the results of <cit.> remain valid here also, with all instances of  replaced by Ω^1-δ_3 in the estimates, given the conclusions of Theorem <ref>. In particular, note that the choice of a(Q,m) is determined by <cit.>, with the proof and the constant ρ_*(Q,m) remaining unchanged herein; namely, a(Q,m) = ρ_*4. Indeed, the compactness procedure therein still yields a limiting area-minimizing current in this setting, since Ω_k converges to zero along the sequence, and thus we may proceed to exploit the monotonicity of mass ratios in the same way. Since such a compactness will be exploited numerous times in the following sections, we elaborate on it in the following remark, which we will refer back to.

 
    We make a note on a procedure that will be often used in the rest of the article. When proving some statements, for instance the crude splitting lemma building up to Proposition <ref> above and Proposition <ref> below, we argue by contradiction. In particular, we consider a sequence of currents T_k, semicalibrated by forms ω_k, and extract a converging subsequence limiting to a certain T_∞, (a valid procedure under our standing hypothesis, e.g. mass bounds on the T_k's). As written, we have no hope to obtain further information on the limit T_∞. However, in all the statement we will prove, we will also have the corresponding Ω_k converging to zero, whence allowing us to deduce that T_∞ is area-minimizing, and proceed as in the relevant proofs of <cit.>. 


The final conclusion of the results in <cit.>, rewritten for a semicalibrated current T is the following, which is the analogue of <cit.>.


Let T, ω and 𝐒=α_1∪⋯∪α_N be as in Assumption <ref>. Then, for every σ, ς>0 there are constants C = C(m,n,Q,δ^*, τ, δ̅)>0 and ε = ε(m,n,Q,δ^*, τ, δ̅, σ, ς)>0 such that the following properties hold.


    
  (i) R∖ B_σ (V) is contained in the outer region R^o.
    
  (ii) There are Lipschitz multi-valued maps u_i : R^o_i →𝒜_Q_i (α_i^⊥) and closed subsets K̅_i (L)⊂ L_i satisfying 
    
    T_L,i𝐩_α_i^-1 (K̅_i(L))= 𝐆_u_i𝐩_α_i^-1 (K̅_i (L))     ∀ L∈𝒢^o  ,

    as well as the estimates <cit.>, and
    
    ∫_R_i |Du_i|^2 ≤ C σ^-2𝐄̂ (T, 𝐒, _4) + C Ω^2-2δ_3  ,

    for R_i := (R∖ B_σ(V))∩α_i.
    
  (iii) If additionally Ω^2-2δ_3≤ε^2 𝐄̂(T, 𝐒, _4) and we set v_i := 𝐄̂ (T, 𝐒, _4)^-1/2 u_i, then there is a map w_i: R_i →𝒜_Q_i (α_i^⊥) which is Dir-minimizing and such that 
    
    d_W^1,2 (v_i, w_i) ≤ς  ,

    where d_W^1,2 is the W^1,2 distance between Q-valued maps; see for instance <cit.> for a definition. 



Note that in order to obtain the conclusion (iii) of Proposition <ref>, we apply <cit.> in place of <cit.>, which is used to establish this conclusion when T is area-minimizing. 



§ CONE BALANCING

In this section, we observe that all of the results of <cit.> remain valid in the case when T is semicalibrated, with all instances of  replaced with Ω^1- δ_3. This is again due to an application of the height bound from Lemma <ref> when proving Proposition <ref> below assuming Proposition <ref>. For the convenience of the reader, and since it will be useful for the succeeding sections, we provide the statements here. First of all, we recall the minimal separation σ() (see (<ref>)) between the planes within a given cone = α_1∪⋯∪α_N as introduced in the preceding section, as well as the maximal separation

    μ()    := max_1≤ i<j ≤ N(α_i∩_1,α_j∩_1)   .



Assume that T and ω are as in Assumption <ref>, 𝐒 = α_1∪⋯∪α_N ∈(Q). Then, there are constants M = M(Q,m,n)>0 and ε_0 = ε_0(Q,m,n)>0 with the following property. Assume that

    Ω^2-2δ_3≤ε_0^2 𝔼 (T, 𝐒, _1) ≤_0^4 ^p(T,_1)  .

 Then there is a subset {i_1, … , i_k}⊂{1, … , N} with k≥ 2 such that, upon setting 𝐒' = α_i_1∪⋯∪α_i_k, the following holds:


  (a) 𝐒' is M-balanced;

  (b) 𝔼 (T, 𝐒', _1) ≤ M 𝔼 (T, 𝐒, _1);

  (c) ^2(∩_1,^'∩_1) ≤ M𝔼(T,,_1);

  (d) M^-1^p (T, _1) ≤μ ()^2 =
μ (^')^2≤ M ^p (T, _1).



The proof of Proposition <ref> reduces to the validity of the following proposition, by exactly the same reasoning as that in <cit.>, given that the crude approximation <cit.> remains valid in this setting with  replaced by Ω^2-2δ_2.


Assume that 
T, ω, and 𝐒 are as in Proposition <ref>. Then there are constants C = C(m,n,N) and ε = ε (m,n,N) with the following property. If we additionally have that

    N≥ 2   and  Ω^2-2δ_3 + 𝔼 (T, 𝐒, _1) ≤ε^2 σ()^2  ,
 
then 𝐒 is C-balanced.


To prove Proposition <ref>, one argues as in <cit.>, namely one first proves a version of the proposition when μ() and σ() are comparable (see <cit.>) and arguing later by induction. The argument follows by contradiction, assuming that the balancing assumption is failing along a sequence of semicalibrated currents T_k with associated semicalibration forms ω_k and cones _k. Note that we have Ω_k^2-2δ_3≤_k^2 σ(_k) with _k ↓ 0. Thus, we still obtain a limiting current T_∞ that is area-minimizing (see Remark <ref> for a similar argument) and supported in some _∞∈(Q)∖ in the non-collapsed case, i.e. when up to subsequence we have lim_k→∞σ(_k) > 0. In light of the main result in <cit.> (see also <cit.>), this contradicts the failure of balancing. Similarly, in the collapsed case lim_k→∞σ(_k) = 0, the compactness argument in <cit.> still yields a limiting Dir-minimizer (the domain of which depends on the case analysis therein), and  thus <cit.> may be applied to contradict the failure of balancing. 



§ ESTIMATES AT THE SPINE


This section is dedicated to the nonconcentration estimates for T near the spine of , much analogous to those in <cit.>, but appearing first in a multiplicity one setting in the seminal work <cit.> of Simon. Our underlying assumption throughout this section will be the following.


Suppose T and ω are as in Assumption <ref> and T (_4) ≤ 4^m (Q+1/2) ω_m. Suppose 𝐒=α_1∪⋯∪α_N is a cone in 𝒞 (Q)∖𝒫 which is M-balanced, where M≥ 1 is a given fixed constant, and let V denote the spine of 𝐒. For a sufficiently small constant = (Q,m,n,M) smaller than the -threshold  in Assumption <ref>, whose choice will be fixed by the statements of Theorem <ref>, Corollary <ref>, and Proposition <ref> below, suppose that

    𝔼 (T, 𝐒, _4) + Ω^2-2δ_3≤ε^2 σ ()^2  .




Assume T, ω, and 𝐒 are as in Assumption <ref>, suppose that Θ (T, 0) ≥ Q, and set r= 1/3√(m-2). Then there is a constant C=C(Q,m,n,M)>0 and a choice of ε = ε(Q,m,n,M)>0 in Assumption <ref> sufficiently small such that

    ∫__r|q^⊥|^2/|q|^m+2  dT (q)
       ≤ C (Ω^2-2δ_3 +  (T, 𝐒, _4)) 
    ∫__r |𝐩_V∘𝐩_T⃗^⊥|^2  dT   ≤ C (Ω^2-2δ_3 +  (T, 𝐒, _4))  .



 This is the analogue of <cit.>. However, note that in order to get the quadratic Ω error improvement in the estimates (<ref>) and (<ref>), we crucially exploit the nature of the error in the first variation of T, and we recall the generalized mean curvature estimate (<ref>). This is very much analogous to the observation made in <cit.> in the case when T is area-minimizing, but since the main term on the right-hand side of (<ref>) and (<ref>) is in terms of the conical excess of T rather than the tilt excess, we provide a sketch proof, for the purpose of clarity.
 

    For any 0 < s < ρ≤ 4, we may test the first variation identity (<ref>) for T with the radial vector field η(| p |τ) p, where τ∈ [s, ρ] and η is a smooth cut-off function that we take to converge to the characteristic function on [0,1], which yields the classical monotonicity formula estimate (see e.g. <cit.>). Now for any 0< r < R <4, let χ∈ C^∞([0, ∞); ℝ) be monotone non-increasing with χ≡ 1 on [0,r] and χ≡ 0 on [R, ∞). Taking s↓ 0, exploiting the hypothesis Θ(T,0) ≥ Q, multiplying by ρ^m and differentiating in ρ, then multiplying by χ (ρ)^2 and integrating over ρ∈[0,R], we obtain the estimate

    ∫__r|q^⊥|^2/|q|^m+2  dT (q)
           ≤ C[∫χ^2 (|q|)  dT (q) - ∑_i Q_i ∫_α_iχ^2 (|q|)  dℋ^m (q)]
            + C ∫__R|Γ (|q|)q^⊥·H⃗_T (q)|/|q|^m   dT (q)  ,

where C=C(m,r)>0 and Γ is a suitable locally bounded function with Γ_L^∞≤C̅ (R^m-r^m) for C̅ = C̅(m). See <cit.> for a more detailed computation. Combining this with the elementary identity ab ≤δ a^22 + b^22δ for δ>0 (to be determined) and the estimate (<ref>), the last term on the right-hand side can then be estimated as follows:
    
    ∫_𝐁_R|Γ(|q|) q^⊥·H⃗_T(q) |1/| q |^m  d‖ T ‖(q)    ≤δ/2∫_𝐁_R|q^⊥|^2/| q |^m+2  d‖ T ‖(q) 
           + C(δ,m) r^m H⃗_T_L^∞^2 ∫_𝐁_R1/| q |^m-2  d‖ T ‖(q) 
       ≤δ/2∫_𝐁_R|q^⊥|^2/| q |^m+2  d‖ T ‖(q) + C(δ,m) Ω^2  .

Inserting this into (<ref>), we obtain

    ∫__r|q^⊥|^2/|q|^m+2  dT (q)
           ≤ C[∫χ^2 (|q|)  dT (q) - ∑_i Q_i ∫_α_iχ^2 (|q|)  dℋ^m (q)]_(I)
           + Cδ∫_𝐁_R|q^⊥|^2/| q |^m+2  d‖ T ‖(q) + C(δ,m) Ω^2  .

Note that this is of the form <cit.> but with ^2 replaced with Ω^2, and the extra term on the right-hand side, which we will eventually absorb into the left-hand side, but due to the domain being larger, this is slightly delicate and needs to be done at the end. The remainder of the proof therein deals with estimating the term (I) above. Observe that the estimate <cit.> may in fact be more precisely written as

    ∫χ^2(|q|)  dT   (q) - ∑_i Q_i ∫_α_iχ^2(|q|)  d^m(q) 
       ≤ C ∫ div_T⃗ X (q)  dT (q) + C ∫__R |x^⊥|^2 dT + ∫χ (|q|) x ·∇_V^⊥χ (|q|)  dS(q) 
            - ∫χ (|q|) 𝐩_T⃗ (x)·∇_V^⊥χ (|q|)  dT (q)
      ,

for the vector field X (q) = χ (|q|)^2 𝐩_V^⊥ (q) and S=(). Thus, in order to follow the arguments of <cit.> verbatim, it merely remains to verify that

    ∫_T⃗ X (q)  dT (q) ≤δ̃∫__R|q^⊥|^2/|q|^m+2  dT(q) + C(δ̃) Ω^2  ,

for δ̃>0 small enough such that, after combining (<ref>) with (<ref>), the first term on the right-hand side of (<ref>) may be reabsorbed into the left-hand side of (<ref>).

To see that (<ref>) holds, we use the first variation of T to write

    ∫ div_T⃗  X   dT = - ∫ X^⊥·H⃗_T   dT  ,

and argue as in (<ref>) to obtain the estimate

    ∫ |X^⊥·H⃗_T |   dT≤δ̃∫__R|q^⊥|^2/|q|^m+2  dT(q) + C(δ̃) H⃗_T _L^∞^2 ∫__R |q|^m+2  dT(q)  ,

from which (<ref>) follows immediately. We may then use the estimates corresponding to those in <cit.>, for which we must use the semicalibrated analogues of the relevant results in <cit.> (with Σ = ^m+n and all occurrences of  replaced with Ω^1-δ_3). The majority of such results were omitted in Section <ref> herein for brevity, but nevertheless hold true.

In summary, when combined with (<ref>), we arrive at the final estimate

    ∫__r|q^⊥|^2/|q|^m+2  dT (q)
           ≤ C(T,,_4) + C(δ,δ̃,m)Ω^2-2δ_3
           + C(δ +δ̃) ∫_𝐁_R|q^⊥|^2/| q |^m+2  d‖ T ‖(q)  .

Invoking, for instance, <cit.>, for δ,δ̃ sufficiently small (depending on Q,m,n), we may indeed absorb the final term on the right-hand side above into the left-hand side, concluding the proof.


    In the proof of Theorem <ref>, one can alternatively notice that the following identity holds, for any vector field X∈ C_c^∞(^m+n∖∂ T;^m+n):
    
    ⟨ dω X, T⃗⟩ = ⟨ dω, X ∧T⃗⟩ = ⟨ dω, X^⊥∧T⃗⟩,

    where the first equality follows by definition of restriction, and where perpendicularity is with respect to the approximate tangent space of T. In particular, this can be used in place of the estimate (<ref>) when establishing (<ref>) and (<ref>) above.


From Theorem <ref>, we further deduce the following, which corresponds to <cit.>.

Assume T, ω, 𝐒 and r are as in Theorem <ref>. Then, there is a choice of ε = ε(Q,m,n,M) in Assumption <ref>, possibly smaller than that in Theorem <ref>, such that for every κ∈ (0,m+2), 

    ∫__r^2 (q, 𝐒)/|q|^m+2-κ  dT (q)
    ≤ C_κ (Ω^2-2δ_3 +  (T, 𝐒, _4))  ,

where here C_κ = C_κ(Q,m,n,M,κ). 


Corollary <ref> follows immediately from the following lemma, combined with Theorem <ref>.


Let T, ω and  be as in Assumption <ref> with 𝐁_1 ⊂Ω and Θ (T, 0)≥ Q. Then we may choose  sufficiently small in Assumption <ref> such that for each κ∈ (0,m+2) we have

    ∫__1 ^2(q, 𝐒)/|q|^m+2-κ  dT (q)
    ≤ C_κ∫__1|q^⊥|^2/|q|^m+2  dT (q)+ C_κ (𝐄̂ (T, 𝐒, _4) + Ω^2)  ,

for some constant C_κ=C_κ(Q,m,n,M,κ)>0.



    Fix κ∈ (0,m+2). As in the proof of Theorem <ref>, we aim to follow the reasoning of the area-minimizing counterpart <cit.> of this lemma, which we may indeed do, provided that we verify

    ∫ div_T⃗ X   dT≤δ∫__1^2(q,)/|q|^m+2-κ  dT(q) + C(δ) Ω^2    ,

for the vector field

    X (q) := ^2 (q,𝐒) (max{r, |q|}^-m-2+κ - 1)_+ q  ,

supported in _1 (and constant on _r). To see the validity of (<ref>), we simply exploit the first variation identity (<ref>) and an estimate analogous to (<ref>) to obtain

    ∫ |X^⊥·H⃗_T|  dT≤δ∫__1 ^2(q,)/|q|^m+2-κ  dT(q) + C(δ) Ω^2 ∫__1^2(q,) |q^⊥|^2  dT(q)   .



Finally we have the following shifted estimates around a given point of high multiplicity (cf. <cit.>).


Assume T, ω, and 𝐒 are as in Assumption <ref> and in addition {Θ (T, ·)≥ Q}∩_ε (0) ≠∅. Then there is a radius r=r(Q,m,n) and a choice of ε = ε(Q,m,n,M) in Assumption <ref>, possibly smaller than those in Theorem <ref> and Corollary <ref>, such that for each κ∈ (0,m+2), there are constants C̅_κ=C̅_κ(Q,m,n,M,κ)>0 and C=C(Q,m,n,M) such that the following holds. If q_0∈_r (0) and Θ (T, q_0)≥ Q, then

    ∫__4r (q_0)^2 (q, q_0 + 𝐒)/|q-q_0|^m+2-κ  dT (q)    ≤C̅_κ (Ω^2-2δ_3 +  (T, 𝐒, _4))  .
    
    |𝐩_α_1^⊥ (q_0)|^2 + μ (𝐒)^2 |𝐩_V^⊥∩α_1 (q_0)|^2    ≤ C (Ω^2-2δ_3 +  (T, 𝐒, _4))  .



The proof of Proposition <ref> is entirely analogous to that of <cit.>, again recalling that the variant of <cit.> herein has Ω^1-δ_3 in place of each instance of .



§ FINAL BLOW-UP AND CONCLUSION

We are now in a position to outline how to conclude the validity of Theorem <ref>, given everything in the preceding sections of this part.

We begin with the following weaker conical excess decay result (see <cit.>), whose validity implies that of Theorem <ref>.

    Fix Q,m,n as before, and let M≥ 1 be as in Proposition <ref>. Fix also ς_1>0. Then, there are constants ε_1 = ε_1(Q,m,n,ς_1)∈ (0,1/2], r^1_1 = r^1_1(Q,m,n,ς_1)∈ (0,1/2] and r^2_1 = r^2_1(Q,m,n,ς_1)∈ (0,1/2], such that the following holds. Suppose that
    
        
  (i) T and Σ are as in Assumption <ref>;
        
  (ii) T(_1)≤ (Q+1/2)ω_m;
        
  (iii) There is ∈𝒞(Q) which is M-balanced, such that
        
    𝔼(T,,_1)≤_1^2σ()^2

        and
        
    __1(ξ)∩{p:Θ(T,p)≥  Q}≠∅    ∀ξ∈ V()∩_1/2  ;

        
  (iv) Ω^2-2δ_3≤_1^2 𝔼 (T, , _1) for every ∈𝒞 (Q).
    
    Then, there is ^'∈𝒞(Q)∖𝒫 such that for some i∈{1,2} we have
    
    𝔼(T,^',_r_1^i) ≤ς_1𝔼(T,,_1)  .


To prove that if <ref> holds, then Theorem <ref> holds, one may proceed exactly as in <cit.>. Indeed, the argument remains unchanged, after replacing  with Ω^1-δ_3 everywhere.

It thus remains to demonstrate that Theorem <ref> holds. With this in mind, we show decay at one of two possible radii as stated therein by considering two possible cases for the cone ∈(Q); collapsed and non-collapsed.


For every Q, m, n and ς_1>0 there are positive constants ε_c = _c(Q,m,n,ς_1)≤ 1/2 and r_c = r_c(Q,m,n,ς_1) ≤ 1/2 with the following property. Assume that 


  (i) T and ω are as in Assumption <ref>, and T (_1) ≤ω_m (Q+1/2);

  (ii) There is a cone 𝐒∈𝒞 (Q) which is M-balanced (with M as in Proposition <ref>), such that (<ref>) and (<ref>) hold with ε_c in place of ε_1, and in addition
μ (𝐒) ≤ε_c;

  (iii) Ω^2-2δ_3≤_c^2𝔼(T,,_1) for every ∈𝒞(Q).

Then, there is another cone 𝐒'∈𝒞 (Q) ∖𝒫 such that 

    𝔼 (T, 𝐒', _r_c)
    ≤ς_1 𝔼 (T, 𝐒, _1)  .




For every Q, m, n, ε^⋆_c >0 and ς_1>0, there are positive constants ε_nc = _nc(Q,m,n,ε^⋆_c,ς_1)≤ 1/2 and r_nc = r_nc(Q,m,n,ε^⋆_c,ς_1) ≤1/2 with the following property. Assume that 


  (i) T and ω are as in Assumption <ref> and T (_1) ≤ω_m (Q+1/2);

  (ii) There is 𝐒∈𝒞 (Q) which is M-balanced (with M as in Proposition <ref>), such that 
(<ref>) and (<ref>) hold with ε_nc in place of ε_1 and in addition
μ (𝐒) ≥ε^⋆_c;

  (iii) Ω^2-2δ_3≤^2_nc𝔼(T,,_1) for every ∈𝒞(Q).

Then, there is another cone 𝐒'∈𝒞 (Q) ∖𝒫 such that 

    𝔼 (T, 𝐒', _r_nc)
    ≤ς_1 𝔼 (T, 𝐒, _1)  .



Since the proofs of Proposition <ref> and Proposition <ref> exploit the results of Sections <ref>-<ref> in the same way as their counterparts in <cit.>, we will merely provide a brief outline of the argument here.

We will argue by contradiction in both the collapsed and the non-collapsed case. 
Namely, we suppose that we have a sequence of currents T_k, corresponding semicalibrations ω_k and cones _k∈(Q) satisfying the hypotheses of either Proposition <ref> with _c^(k) = 1k→ 0 or Proposition <ref> for some fixed _c^⋆>0 but _nc^(k) = 1k→ 0, but such that the respective decay conclusions (<ref>), (<ref>) fail for any possible choice of radii r_c,r_nc.

In particular,

    (Ω^2-2δ_3/𝔼 (T_k, 𝐒_k, _1) + 𝔼 (T_k, 𝐒_k, _1)/σ (𝐒_k)^2)→ 0   .


For the non-collapsed case, we recall the coherent outer approximations u_i^k of Proposition <ref>. Meanwhile, in the collapsed case, we introduce transverse coherent approximations as in <cit.> as follows. Up to extracting a subsequence, we may write _k=α_1∪⋯∪α_N^k for α_1 and N fixed, and write each plane α_i^k as the graph of a linear map A_i^k:α_1 →α_1^⊥. We may in addition reparameterize u_i^k over α_1 to obtain a map v_i^k : R^o_1 →𝒜_Q_i^k (α_1^⊥) whose graph coincides with that of u_i^k over 𝐩_α_1^-1 (R_1^o), where R^o_1 is a suitable graphicality region, defined rigorously in <cit.>. The transverse coherent approximations are then defined to be the collection of maps

    w_i^k := v_i^k⊖ A_i^k: R̃_1^o →_Q_i^k(α_1^⊥)  ,      i=1,…, N  ,

where we use the notational shorthand g⊖ f for the multivalued map ∑_i  g_i - f. Observe that the estimates of <cit.>, with _k replaced by Ω_k^1-δ_3, are valid for A_i^k, v_i^k and w_i^k. Furthermore, the nonconcentration estimates <cit.> hold, again with Ω_k^1-δ_3 in place of _k in the errors.

We in turn define the normalizations

    u̅^k_i    := u^k_i/√(𝔼 (T_k, 𝐒_k, _1))
    w̅^k_i    := w^k_i/√(𝔼 (T_k, 𝐒_k, _1))  ,

on (_r̅∩α^k_i) ∖ B_1/k (V) and (_r̅∩α_1)∖ B_1/k (V) respectively, for r̅ := r4 with r=r(Q,m,n) as in Proposition <ref> (which we may assume is contained in the domains of definition for u_i^k and w_i^k). Arguing as in <cit.>, we may then assume that, up to subsequence, u̅^k_i and w̅^k_i converge strongly in W^1,2 locally away from V and strongly in L^2 on the entirety of B_r to W^1,2 maps u̅_i and w̅_i that are Dir-minimizing on (_r̅∩α^k_i) ∖ V and (_r̅∩α_1)∖ V, and since V has capacity zero, they are in fact Dir-minimizing on (_r̅∩α^k_i) and (_r̅∩α_1) respectively. Moreover, the conclusions of <cit.> are satisfied. Exploiting <cit.> and proceeding as in <cit.> to propagate this decay (up to subtracting an appropriate superposition of linear maps), we arrive at the desired contradiction.



PART: 
Failure of monotonicity for semicalibrated intrinsic planar frequency

The aim of this last part is to provide a simple example of a semicalibrated current with good decay properties towards a flat tangent plane, but that does not exhibit an almost monotone planar frequency function as introduced in <cit.>. This therefore illustrates a difference between the present approach to Theorem <ref> when compared to trying to adapt the corresponding one in <cit.>, which we have been unable to adapt to the semicalibrated setting. En route to this, we will point out some differences between the setting herein and the area-minimizing one. 



§ PLANAR FREQUENCY FUNCTION

We begin by introducing the analogue of the intrinsic planar frequency function of <cit.> for a semicalibrated current T. Let z ∈ℝ^n + m, let π⊂^m+n be an m-dimensional plane and let ρ_0 > 0. 

We will henceforth work under the following assumption.


   Let ρ_0>0. Suppose that T is a semicalibrated rectifiable current in 𝐂_ρ_0(z, π) satisfying 

    ∂ T 𝐂_ρ_0(z, π) = 0,   and  sup_p ∈ T ∩𝐂_ρ_0(z, π)(p, π + z) < ∞.



Let ϕ [0, ∞) → [0,1] be the monotone Lipschitz cutoff function defined in (<ref>). For r ∈ (0, ρ_0], we can define the intrinsic L^2 height of T at scale r around z with respect to the plane π to be

    H_T, π, z(r) := 2/r^m-1∫_𝐂_r(z, π) ∖𝐂_r/2(z, π)^2(p, π + z) |∇_T⃗ |𝐩_π(p-z)| |^2/|𝐩_π(p-z)|  d‖ T ‖(p)  ,

Note that (<ref>) can be rewritten as 

    H_T, π, z(r) = - 1/r^m-1∫^2(p, π + z) |∇_T⃗ |𝐩_π(p-z)| |^2/ |𝐩_π(p-z)|ϕ^'(|𝐩_π(p-z)|/r)   d‖ T ‖(p) .


Furthermore, we can introduce the intrinsic Dirichlet energy of T at scale r around z with respect to the plane π as

    D_T, π, z(r) := 1/2r^m-2∫|𝐩_T(p) - 𝐩_π (p) |^2 ϕ(|𝐩_π(p-z)|/r)   d ‖ T ‖(p) .

Note that D_T,π,z(r) may be considered as a regularization of the non-oriented tilt excess of T in the cylinder _r(z,π), cf. (<ref>). Pertinent to our setting, we define the corresponding intrinsic semicalibrated term

    L_T,π,z(r):= 1/2r^m-2∫ T(d ω_π^⊥(p))ϕ(|𝐩_π(p-z)|/r)   d ‖ T ‖(p)

and

    Γ_T,π,z(r) := D_T,π,z(r) + L_T,π,z(r).

Notice that the additional term T(d ω_π^⊥(p)) above arises from testing the first variation of T with the variation vector field  X(p)=ϕ(|𝐩_π(p-z)|r)_π^⊥(p); see <cit.>. In analogy with <cit.>, whenever H_T, π, z(r) > 0, we may define the intrinsic planar frequency function N_T, π, z(r) of T at Z relative to P by 

    N_T, π, z(r) = Γ_T, π, z(r)/H_T, π, z(r)  .

Note that when T is calibrated (and in particular area-minimizing), i.e. when dω≡ 0, the above frequency indeed reduces to the one introduced in <cit.> for area-minimizing currents. In <cit.>, it is shown that the intrinsic planar frequency associated to area-minimizing integral currents is almost-monotone under a suitable decay hypothesis (see Section <ref> below for a more precise statement), thus laying the foundation for a more refined analysis of the singular set of area-minimizers. 

The main result of this part is the following.

    There exists a smooth, radially symmetric function f: B_1 ⊂π≡ℝ^m ×{0}→ such that the associated semicalibrated current _f has the property
    
    N_T, π, 0(r) → + ∞     as r ↓ 0.



Namely, we provide the construction of an example that not only violates almost-monotonicity of the intrinsic planar frequency function in the semicalibrated setting, but allows for it to diverge to +∞ as the scale goes to zero. Before we proceed with the construction, let us provide a more detailed heuristic explanation, together with a comparison with what happens in the area-minimizing setting.



 §.§ Comparison with Proposition <ref>

One may consider the statement of Proposition <ref> as a conclusion that a posteriori, one does not require center manifolds to take blow ups at points x∈_Q(T) with singularity degree (T,x) ∈ [1,2-δ_2) (and indeed, this is the case in the work <cit.> when the planar frequency is in [1,2)). This suggests that one may possibly use a planar frequency function like N_T,π,z (in place of a frequency relative to center manifolds) to analyze such points. This is not inconsistent with the validity of Theorem <ref>. Indeed, in the latter, the smooth submanifold (f) coincides with the center manifold associated to T=_f locally around the origin. On the other hand, our choice of f will have infinite order of vanishing at the origin, which corresponds to blow-up of the planar frequency function there. Note, however, that 0 is not a flat singular point of _f in this case; in fact _f has no singularities. For the same reason, no single-sheeted example will be inconsistent with the validity of Proposition <ref>. However, since it is not clear how to meaningfully restrict the notion of intrinsic planar frequency to such a specific scenario in order to improve its properties, we do not pursue this any further.



 §.§ Comparison with <cit.>

The area-minimizing hypothesis is crucially used in <cit.> to control the error terms arising when differentiating H_T, π, z, and D_T, π, z, which in turn produce errors for the radial derivative of N_T,π,z. More precisely, the area-minimizing property of T is exploited therein to infer the following bounds, cf. Lemma 3.9 and Lemma 3.11 of loc. cit., 

    | H_T, π, z^'(r) + 2 r^- m∫|∇^⊥ |𝐩_π(p-z)| |^2 |𝐩_π(p-z)| ϕ'(|𝐩_π(p-z)|/r)   d‖ T ‖(p) |
               ≤ C η^2γ r^2 αγ - 1 H_T, π, z(r)  ,

where ∇^⊥≡∇_T^⊥, and 

    | D_T, π, z^'(r) +  2 r^- m∫|_π^⊥(∇_T |𝐩_π(p-z)|) |^2/|∇_T |𝐩_π(p-z)| |^2 |𝐩_π(p-z)| ϕ'(|𝐩_π(p-z)|/r)   d‖ T ‖(p) |
                   ≤C/rD_T,π,z(r)^γ((m - 1)D_T, π, z(r) + r D_T, π, z^'(r)),

for some γ=γ(Q,m,n)>0 and C=C(Q,m,n)>0, whenever T as in Assumption <ref> satisfies the mass ratio bounds 

    Θ(T,z) ≥ Q,     T(_7ρ_0/4(z,π)) ≤ (Q+δ) ω_m (7ρ_0/4)^m  ,

for some δ=δ(Q,m,n) > 0 and the additional planar decay hypothesis

    1/ω_m (7s /4)^m+2∫__7s /4(z,π)^2(p, π +z)   dT(p) ≤η^2 (s/ρ_0)^2α    ∀ s ∈ [σ_0,ρ_0]  ,

for some σ_0∈ (0,ρ_0), some η_0=η_0(Q,m,n)>0 and η∈ (0,η_0].

The estimates (<ref>) and (<ref>), together with the variational identities for H_T,π,z, D_T,π,z, in turn can be used to prove the almost-monotonicity 

    N_T,π,z(r) ≤ e^Cη^γ (s/ρ_0)^αγ N_T,π,z(s)    ∀σ_0 ≤ r < s ≤ρ_0 ,

of the intrinsic planar frequency function in the area-minimizing case, provided that H_T,π,z(τ) > 0 on [r,s]. The first challenge when trying to adapt this argument to the semicalibrated setting is that the terms H_T, π, z^' and D_T, π, z^' will now contain extra errors depending on the semicalibration dω (due the fact that the first variation of a semicalibrated current has a non-vanishing right-hand side). Moreover, one must additionally consider the behavior of L_T, π, z^' when controlling the variational error terms. Complications arise when one tries to bound all of these error terms by powers of the intrinsic Dirichlet energy and L^2 height, as in (<ref>) and (<ref>). 



§ PROOF OF THEOREM <REF>

We are now in a position to construct a counterexample to the almost-monotonicity of the intrinsic planar frequency N_T,π,0 associated to a semicalibrated current T=_f associated to the graph of a smooth radially symmetric function f relative to the plane π≡^m×{0}⊂^m+1, as claimed in Theorem <ref>. 

To this end, consider f ∈ C^∞(B_1), where B_1 ⊂^m ×{0}⊂^m+1. Consider then the graph of f as a submanifold of ℝ^m + 1: 

    (f) = {(x, f(x));   x ∈ B_1}.

We claim that (f) is a semicalibrated submanifold of ℝ^m +1. Indeed, consider the unit normal to (f) given by 

    ν_x = 1/√(1 + |∇ f(x) |^2)(- ∇ f(x), 1),

and define the m-form ω(X_1, …, X_m) = (X_1, …, X_m, ν) for vectors X_1, …, X_m. It is then straightforward to check that |ω(Z_1, …, Z_m) |≤ 1 for unit vectors Z_1, …, Z_m. Furthermore, we have that ω(Y_1, …, Y_m) = 1, if Y_i are orthonormal vectors belonging to T_(x, f(x))(f), so that the submanifold (f) ⊂^m+1 is semicalibrated. In particular, the current T=_f associated to it (see e.g. <cit.>) is also semicalibrated. From this, it is clear that not every semicalibrated current can have an almost-monotone planar frequency function. We will however provide a natural example to illustrate the blow-up of planar frequency. We recall the quantities D_T, π, 0(r), H_T, π, 0(r), and L_T,π,z(r) introduced in the preceding section for this particular choice of T, with π≡ℝ^m×{0}. 



 §.§ Intrinsic quantities for a graph

We start by unpacking the definitions of intrinsic Dirichlet energy, L^2 height, and the semicalibrated term in the case where T=_f for f as above; we will define f later. Let us begin with the energy D_T, π, 0(r). The projection matrix associated with π is given by 

    _π = [ _m × m      0;      0      0 ],

while the projection matrix for the tangent space T_(x, f(x))(f) is 

    _T_(x, f(x))(f)    = _(m + 1) × (m + 1) - ν_x ⊗ν_x
 
In particular, we can compute 

    1/2|_T_(x, f(x))(f) - _π|^2 = 1- (_T_(x, f(x))(f) : _π) = |∇ f |^2/1 + |∇ f |^2,

where A:B denotes the Hilbert-Schmidt inner product between matrices A,B, so that 

    D_T, π, 0(r)     = r^2 - m∫ϕ(|x|/r) |∇ f(x) |^2/1 + |∇ f(x) |^2  d ‖ T‖(x,f(x))  .

Letting ϕ converge to the characteristic function of the unit interval from below, and recalling the area formula for a graph, we obtain

    D_T, π, 0(r) = r^2 - m∫_B_r|∇ f |^2/√(1 + |∇ f |^2)  dℒ^m.

We can now turn to the height H_T, π, 0(r). Write 

    H_T, π, 0(r) = - r^1 - m∫| f(x)|^2 |∇_T |x| |^2/|x|ϕ'(|x|/r)   d ‖ T ‖(x,f(x)),

and, after recalling ∇ r = (x, 0)/|x|, we can compute 

    |∇_T |x| |^2 = |_T_(x, f(x))(f) (∇ |x|) |^2 = 1 + |∇_θ f |^2/1 + |∇ f |^2,

where ∇_θ denotes the angular part of the gradient. Thus, after letting ϕ converge to the characteristic function of the unit interval, we infer 

    H_T, π, 0(r) = r^1 - m∫_∂ B_ρ(0)| f |^2 1 + |∇_θ f |^2/√(1 + |∇ f |^2)  d ℋ^m - 1.

Finally, we rewrite the definition of the semicalibrated term 

    L_T,π,0(r) = 1/2r^m-2∫⟨ dω (0,y), T⃗⟩ϕ(|x|/r)   d‖ T‖(x,y),

where we write ^m+1∋ p=(x,y)∈π×π^⊥. Letting ϕ converge to the characteristic function of the unit interval and again using the area formula, we obtain  

    L_T,π,0(r) = 1/2r^m-2∫_B_r⟨ dω (0, …, 0, f), T⃗⟩√(1 + |∇ f |^2)  dℒ^m  ,

where T⃗(x) is the m-vector τ_1 ∧…∧τ_m, for {τ_i}_i a basis of the tangent space T_(x, f(x))(f). Note that here we have also used that in this particular setting, _π^⊥(x,y) = (0, …, 0, f(x)). Note that 
    τ_i = e_i + (∇ f · e_i) e_m + 1/√(1 + (∇ f · e_i)^2),
 for i ∈{1, …, m}, where {e_i}_i is a basis of ℝ^m + 1. The (m +1)-form dω is given by

    dω = (-1)^m( ∇ f/√(1 + |∇ f |^2)) dx_1 ∧ dx_2 ∧…∧ dx_m +1  .

Note that one can alternatively use Cartan's magic formula to arrive at the same final expression for dω. Then, 

    (0, …, 0, f) ∧T⃗ = (f e_m + 1) ∧τ_1 ∧…∧τ_m = (- 1)^m f/Π_i = 1^m√(1 + (∇ f · e_i)^2) e_1 ∧ e_2 ∧…∧ e_m + 1,

so that 

    ⟨ dω (0, …, 0, f), T⃗⟩    =( ∇ f/√(1 + |∇ f |^2)) f/Π_i = 1^m√(1 + (∇ f · e_i)^2)

Thus, the semicalibrated term is 

    L_T,π,0(r) = 1/2r^m-2∫_B_r( ∇ f/√(1 + |∇ f |^2)) f/Π_i = 1^m√(1 + (∇ f · e_i)^2)√(1 + |∇ f |^2)  dℒ^m  .




 §.§ Definition of f

We are now in a position to define our radially symmetric function. Let f given by 

    f(x) = 
                e^-1/| x |^2    x ≠ 0 
    
                0     x=0  .

Note that ∇ f(0) = 0 and that f is indeed radially symmetric, so that introducing polar coordinates we can write (abusing notation) f(r) = e^-1/r^2, for r ≥ 0. By the argument at the beginning of this section, the current T=_f associated to the graph of this function is semicalibrated. Furthermore, note that the hypothesis of <cit.> are satisfied. More precisely, let ρ_0 > 0, and consider T𝐂_7ρ_0/4(0, π) for π=^m×{0}⊂^m+1. In particular, Θ(T, 0) = 1, and the almost-monotonicity of mass ratios (see e.g. <cit.>) guarantees that ‖ T ‖(𝐂_7ρ_0/4(0, π)) ≤ (1 + δ)ω_m (7ρ_0/4)^m for ρ_0 sufficiently small, where δ is the parameter of <cit.>. In addition, thanks to the exponential decay of f towards 0, there exist η > 0, σ_0 ∈ (0, ρ_0), and α∈ (0, 1) such that the decay hypothesis (<ref>) holds about the origin, namely

    1/ω_m (7s/4)^m + 2∫_𝐂_7s/4(0, π)^2(p, π)   d‖ T ‖(p) ≤η^2 ( s/ρ_0)^2 α,

for all ρ∈ [σ_0, ρ_0].
Consider now the planar frequency function N_T,π,0 and use (<ref>) and (<ref>) to write the energy and the height for this particular function:

    D_T, π, 0(ρ) = ρ^2 - mω_m - 1∫_0^ρ4e^-2/r^2 r^-6/√(1 + 4e^-2/r^2 r^-6) r^m - 1  dr  ,

and 

    H_T, π, 0(ρ) =  ω_m - 1e^-2/ρ^2/√(1 + 4e^-2/ρ^2ρ^-6)  .

Note that H_T, π, 0(ρ) > 0 for all ρ > 0, implying that N_T, π, 0 is always well-defined. Thus, for any ρ>0 sufficiently small, the classical planar frequency function can be estimated from below as follows: 

    D_T, π, 0(ρ)/H_T, π, 0(ρ)    = e^2/ρ^2√(1 + 4e^-2/ρ^2ρ^-6)ρ^2 - m∫_0^ρ4e^-2/r^2 r^-6/√(1 + 4e^-2/r^2 r^-6) r^m - 1  dr 
        = ρ^-1 - m e^1/ρ^2√(e^2/ρ^2ρ^6 + 4)∫_0^ρ4e^-1/r^2 r^-3/√(4 + e^2/r^2 r^6) r^m - 1  dr 
       ≥ C ρ^2 - m e^2/ρ^2∫_0^ρ e^- 2/r^2r^m - 7  dr 
       ≥ C ρ^2 - m e^2/ρ^2 2^m/2 - 4Γ(3 - m/2, 2/ρ^2),

where Γ(s, x) is the incomplete gamma function 

    Γ(s, x) = ∫_x^∞ t^s - 1 e^- t  dt.

Recalling now the asymptotic Γ(s, x)x^- s + 1e^x→ 1, as x →∞, we deduce that 

    N_T, π, 0(ρ) ≥C/ρ^2( Γ(3 - m/2, 2/ρ^2) e^2/ρ^2( 2/ρ^2)^1 + m/2 - 3) =: 1/ρ^2η(ρ),

where η(ρ) → 1 as ρ→ 0, which yields

    D_T, π, 0(ρ)/H_T, π, 0(ρ)→∞    as ρ→ 0  ,

as desired. We now wish to compute the semicalibrated term in the intrinsic planar frequency, namely L_T,π,0(ρ)/H_T, π, 0(ρ). We record the minimal surface equation for a radial function on ℝ^m:

    ( ∇ f/√(1 + |∇ f |^2)) = 1/√(1 + (f^')^2)( f^''/1 + (f^')^2 + m-1/rf^').

Thus, we can estimate

    L_T,P,0(ρ)/H_T, P, 0(ρ)   = e^1/ρ^2/ρ^m+1√(e^2/ρ^2ρ^6 + 4)∫_0^ρr^m - 1e^- 2/r^2√(1+(f'(r))^2)/Π_i = 1^m√(1 + (∇ f · e_i)^2)[2(m-1)/r^4 + 4 - 6r^2/r^6 + 4 e^-2/r^2]     dr 
         ≥C e^1/ρ^2/ρ^m+1√(e^2/ρ^2ρ^6 + 4)∫_0^ρ r^m - 1e^- 2/r^2[2(m-1)/r^4 + 4 - 6r^2/r^6 + 4 e^-2/r^2]   dr.

Splitting then the square bracket and analyzing the two integrands separately via the incomplete Gamma function again, one deduces that L_T,π,0(ρ)/H_T, π, 0(ρ) also diverges as ρ→ 0. 


amsalpha 
 

