# Lower bounds on mapping content and quantitative factorization through trees

Guy C. David  Department of Mathematical Sciences
Ball State University, Muncie, IN 47306<EMAIL_ADDRESS>and  Raanan Schul
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651<EMAIL_ADDRESS>
###### Abstract.

We give a simple quantitative condition, involving the “mapping content” of
Azzam–Schul, that implies that a Lipschitz map from a Euclidean space to a
metric space must be close to factoring through a tree. Using results of
Azzam–Schul and the present authors, this gives simple checkable conditions
for a Lipschitz map to have a large piece of its domain on which it behaves
like an orthogonal projection. The proof involves new lower bounds and
continuity statements for mapping content, and relies on a “qualitative”
version of the main theorem recently proven by Esmayli–Hajłasz.

###### 2010 Mathematics Subject Classification:

28A75, 53C23, 30L99.

G. C. David was partially supported by the National Science Foundation under
Grants No. DMS-1758709 and DMS-2054004. R. Schul was partially supported by
the National Science Foundation under Grant No. DMS-1763973.

##  1\. Introduction

The goal of this paper is to explain a link between two questions that one can
ask about a Lipschitz mapping from a Euclidean space into a metric space. We
first describe these questions philosophically, and then give precise details.

The first question is: Are there suitable coordinates on which $f$ looks like
a simple orthogonal projection? From a geometric perspective, the nice
properties of such a projection are that the fibers of the mapping are
parallel and are parametrized in a uniform way by Euclidean spaces of the
expected dimension.

As one cannot expect every Lipschitz map to look like a projection in
coordinates, we ask instead:

######  Question 1.1.

Are there large pieces of the domain on which the restriction of $f$ admits
such a change of coordinates?

In other words, we are looking for a large piece of the domain on which we
have a (non-smooth) implicit function theorem. This question was explored in
detail recently in [1, 10, 5], as we make precise and explain further below.

It turns out that there are highly non-trivial mappings – first discovered by
Kaufman [13] – that fail to admit any such pieces at all. Kaufman’s example is
a Lipschitz map of $[0,1]^{3}$ onto $[0,1]^{2}$ that nonetheless fails in a
precise way to behave like an orthogonal projection
$\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$ on any set of positive measure.

Kaufman’s example provides the link to our second question, because it factors
through a tree.

######  Definition 1.2.

A metric tree is a geodesic metric space in which every geodesic triangle is
isometric to a tripod.

We say that a Lipschitz map $f\colon X\rightarrow Y$ factors through a tree if
there is a metric tree $T$ and two Lipschitz maps $g\colon X\rightarrow T$ and
$h\colon T\rightarrow Y$ such that $f=h\circ g$.

If the two maps $h$ and $g$ are both $L$-Lipschitz, we say that $f$ factors
through a tree by $L$-Lipschitz maps.

Thus, the second question we ask about a given Lipschitz map $f$ is:

######  Question 1.3.

How can we tell whether $f$ factors through a tree, or (taking the
quantitative viewpoint) is close to factoring through a tree?

Our initial motivation for this question was to understand whether Kaufman-
type examples are the only ones that can provide negative answers to Question
1.1 in low-dimensional situations.

Maps that factor through trees are degenerate in some topological ways. As a
simple illustration, the restriction of such a map to any Jordan curve in the
domain must fail to be injective. Nonetheless, such mappings can exhibit
unexpected geometric complexity, as shown by Kaufman’s example, whose behavior
is quite surprising in view of the one-dimensionality of trees. Further
examples of mappings that factor through trees and nonetheless have “large”
images appear in [1, Section 2].

It turns out that the answers to both Questions 1.1 and 1.3 are governed by
the same quantity, the so-called “$(n,m)$-mapping content” first introduced in
[1].

######  Definition 1.4.

Let $n,m$ be non-negative integers and let $E\subseteq Q_{0}=[0,1]^{n+m}$ be a
set. We define the “$(n,m)$-mapping content” of $f$ on $E$ as

${\mathcal{H}}^{n,m}_{\infty}(f,E):=\inf\sum_{Q_{i}}{\mathcal{H}}^{n}_{\infty}(f(Q_{i}))\textnormal{side}(Q_{i})^{m},$

where the infimum is taken over all coverings $\\{Q_{i}\\}$ of $E$ by dyadic
cubes with disjoint interiors in $Q_{0}$.

Here ${\mathcal{H}}^{n}_{\infty}$ refers to the $n$-dimensional Hausdorff
content, defined in subsection 2.3.

######  Remark 1.5.

We note that this definition of ${\mathcal{H}}^{n,m}_{\infty}$ is precisely
that used in [5]. It differs slightly from that stated in [1, Equation (1.3)]
in that the infimum is over coverings by dyadic cubes rather than arbitrary
cubes, although the results in [1] really only require dyadic cubes. The two
definitions are weakly comparable, in the sense that if one quantity is small
than the other is small; this follows from [5, Corollary E].

The quantity ${\mathcal{H}}^{n,m}_{\infty}$ serves in some sense as a “coarse”
(and metric) analog of the $L^{1}$-norm of the $n$-dimensional Jacobian of
$f$, defined as the product of the $n$ largest singular values of the
derivative of $f$. Thus, a mapping with small $\mathcal{H}^{n,m}_{\infty}$
might be viewed as highly compressing $n$-dimensional volumes on some scales.
This must be interpreted with care, however, given Kaufman’s example, which
has $\mathcal{H}^{2,1}_{\infty}=0$ but maps onto the unit square $[0,1]^{2}$.
This is discussed further in [5, Section 3].

When $n=2$, the present authors made the following conjecture in [5,
Conjecture 1.13]:

######  Conjecture 1.6 ([5]).

Let $Q_{0}=[0,1]^{2+m}$ and let $f\colon Q_{0}\rightarrow\ell^{\infty}$ be a
$1$-Lipschitz mapping.

  * •

(Qualitative version) If $\mathcal{H}^{2,m}_{\infty}(f,Q_{0})=0$, then $f$
factors through a tree.

  * •

(Quantitative version) For every $\epsilon>0$, there is a
$\delta=\delta(\epsilon,m)$ with the following property: If
$\mathcal{H}^{2,m}_{\infty}(f,Q_{0})<\delta$, then there is a $1$-Lipschitz
map $g\colon Q_{0}\rightarrow\ell^{\infty}$ that factors through a tree and
satisfies $\sup_{Q_{0}}\|g-f\|_{\infty}<\epsilon$.

This conjecture was inspired by the techniques of [16] and the evidence of
Kaufman’s example, which provides a negative answer to Question 1.1, factors
through a tree, and has $\mathcal{H}^{2,1}_{\infty}$ equal to zero. The same
properties hold for the examples in [1, Section 2]. Essentially, the
conjecture asks whether Kaufman-type examples are the only examples of
mappings with small mapping content, and hence mappings providing negative
answers to Question 1.1, in the case $n=2$.

######  Remark 1.7.

In the case $n\leq 1$ or $m=0$, mappings from $[0,1]^{n+m}$ with vanishing or
small $\mathcal{H}^{n,m}_{\infty}$ are easy to characterize, while in the case
$n>2$ the problem appears difficult due to the constructions of topologically
non-trivial low-rank Lipschitz mappings in [16, 9]. Thus, the case $n=2$ that
we address here is simply the first remaining open problem in a long list.
This is discussed further in [5, Section 1.4].

Recently, Esmayli–Hajłasz [6] showed that a mapping with
$\mathcal{H}^{2,m}_{\infty}$ equal to zero must factor through a tree,
answering the qualitative cases of Question 1.3 and, equivalently, Conjecture
1.6. (This is the equivalence “(e) $\Leftrightarrow$ (a)” in [6, Theorem 1.1],
combined with their Remark 1.2.)

######  Theorem 1.8 (Esmayli–Hajłasz [6]).

Let $Q_{0}=[0,1]^{2+m}$ and $f\colon Q_{0}\rightarrow Y$ be a Lipschitz map
into a metric space. Then $\mathcal{H}^{2,m}_{\infty}(f,Q_{0})=0$ if and only
if $f$ factors through a tree. Moreover, in this case, if $f$ is $1$-Lipschitz
then $f$ factors through a tree by $1$-Lipschitz maps.

Motivated by the more quantitative concerns of [1, 5], we use Theorem 1.8 to
prove a quantitative stability theorem: maps with small, but not necessarily
vanishing, mapping content must be close to factoring through trees.

Our main theorem in this paper is the following, completely resolving Question
1.3 and finishing the proof of Conjecture 1.6.

######  Theorem A.

Let $m$ be a non-negative integer. For every $\epsilon>0$, there is a
$\delta=\delta(\epsilon,m)>0$ with the following property: Let $f\colon
Q_{0}:=[0,1]^{2+m}\rightarrow\ell^{\infty}$ be a $1$-Lipschitz map. If
$\mathcal{H}^{2,m}_{\infty}(f,Q_{0})<\delta$, then there is a $1$-Lipschitz
map $g\colon Q_{0}\rightarrow\ell^{\infty}$ such that

$\sup_{x\in Q_{0}}\|g(x)-f(x)\|_{\infty}<\epsilon$

and $g$ factors through a tree by $1$-Lipschitz maps.

Conversely, for every $\epsilon^{\prime}>0$, there is a
$\delta^{\prime}=\delta(\epsilon^{\prime},m)>0$ such that if $f,g\colon
Q_{0}:=[0,1]^{2+m}\rightarrow\ell^{\infty}$ are $1$-Lipschitz maps, $g$
factors through a tree by $1$-Lipschitz maps, and

$\sup_{x\in Q_{0}}\|g(x)-f(x)\|_{\infty}<\delta^{\prime},$

then $\mathcal{H}^{2,m}_{\infty}(f,Q_{0})<\epsilon^{\prime}$.

######  Remark 1.9.

As every separable metric space embeds isometrically in $\ell^{\infty}$, the
restriction in Conjecture 1.6 and Theorem A that the target space is
$\ell^{\infty}$ is no restriction at all; it simply allows for a cleaner
statement.

Without the restriction to $n=2$ in Theorem A, the result is false, as alluded
to in Remark 1.7. On the other hand, the assumption $n=2$ enters this paper
only through Theorem 1.8, so our methods in fact show the following in all
dimensions.

######  Theorem B.

For every $n,m$ and $\epsilon>0$, there is a $\delta=\delta(\epsilon,n,m)>0$
with the following property: If $f\colon
Q_{0}:=[0,1]^{n+m}\rightarrow\ell^{\infty}$ is a $1$-Lipschitz map and
$\mathcal{H}^{n,m}_{\infty}(f,Q_{0})<\delta$, then there is a $1$-Lipschitz
map $g\colon Q_{0}\rightarrow\ell^{\infty}$ such that

$\sup_{x\in Q_{0}}\|g(x)-f(x)\|_{\infty}<\epsilon$

and $\mathcal{H}^{n,m}_{\infty}(g,Q_{0})=0$.

Conversely, for every $\epsilon^{\prime}>0$, there is a
$\delta^{\prime}=\delta(\epsilon^{\prime},n,m)>0$ such that if $f,g\colon
Q_{0}:=[0,1]^{n+m}\rightarrow\ell^{\infty}$ are $1$-Lipschitz maps,
$\mathcal{H}^{n,m}_{\infty}(g,Q_{0})=0$, and

$\sup_{x\in Q_{0}}\|g(x)-f(x)\|_{\infty}<\delta^{\prime},$

then $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})<\epsilon^{\prime}$.

Thus, attempts at higher-dimensional generalizations of Theorem A could
perhaps first proceed by attempting to characterize maps with vanishing
mapping content in higher dimensions. Progress on this appears in [7,
Corollary 7.22].

Before proceeding with more precise motivation and a description of the
remaining results of the paper, we explicitly state a simple corollary of
Theorem A which does not require the notion of mapping content. This is just
the case $m=0$, in which case $\mathcal{H}^{2,m}_{\infty}(f,Q_{0})$ is simply
$\mathcal{H}^{2}_{\infty}(f(Q_{0}))$, the standard $2$-dimensional Hausdorff
content of the image of $f$. Thus:

######  Corollary 1.10.

For every $\epsilon>0$, there is a $\delta=\delta(\epsilon)>0$ with the
following property: Let $f\colon Q_{0}:=[0,1]^{2}\rightarrow\ell^{\infty}$ be
a $1$-Lipschitz map. If $\mathcal{H}^{2}_{\infty}(f(Q_{0}))<\delta$, then
there is a map $g\colon Q_{0}\rightarrow\ell^{\infty}$ such that

$\sup_{x\in Q_{0}}\|g(x)-f(x)\|_{\infty}<\epsilon$

and $g$ factors through a tree.

In other words, a $1$-Lipschitz map from a square can only compress
$2$-dimensional content insofar as it is close to factoring through a tree.

###  1.1. Motivation from geometric measure theory: “Hard Sard sets”

As motivation, we now explain more precisely the connection between our
results and Question 1.1.

A classical result in geometric measure theory says that for a Lipschitz map
$f:\mathbb{R}^{n+m}\to\mathbb{R}^{n}$, the pre-image of
${\mathcal{H}}^{n}$–almost every point of $f(\mathbb{R}^{n+m})$ is
$m$-rectifiable. (See, e.g., [8, Theorem 3.2.22].) That is, almost every fiber
of $f$ can be covered by Lipschitz images of $m$-dimensional Euclidean space,
up to $m$-dimensional measure zero.

As with many problems in geometric measure theory, there is a natural
quantitative follow-up question: is it possible to parametrize these fibers
(or at least a large fraction of them) in a uniform way? More specifically, is
it possible to rearrange the domain (or at least a large fraction of it) so
that the fibers are parallel $m$-planes?

The canonical example of a map whose fibers are parallel $m$-planes is a
simple orthogonal projection, which connects this to Question 1.1. The recent
works [1, 10, 5] all explore this question. We first make more precise, with a
definition from [1, 5], what we mean by “straightening out” or “making
parallel” the fibers of a mapping.

######  Definition 1.11.

Let $Q_{0}=[0,1]^{n+m}$ and $f\colon Q_{0}\rightarrow Y$ be a Lipschitz map
into a metric space. Let $E\subseteq Q_{0}=[0,1]^{n+m}$ and
$g:E\rightarrow\mathbb{R}^{n+m}$ be a bi-Lipschitz mapping. We call $(E,g)$ a
Hard Sard pair for $f$ if there is a constant $C_{Lip}$ such that the
following conditions hold.

Write points of $\mathbb{R}^{n+m}$ as $(x,y)$ with $x\in\mathbb{R}^{n}$ and
$y\in\mathbb{R}^{m}$. Let $F=f\circ g^{-1}$. We ask that:

  1. (i)

$g$ extends to a globally $C_{Lip}$-bi-Lipschitz homeomorphism from
$\mathbb{R}^{n+m}$ to $\mathbb{R}^{n+m}$.

  2. (ii)

If $(x,y)$ and $(x^{\prime},y^{\prime})$ are in $g(E)$, then
$F(x,y)=F(x^{\prime},y^{\prime})$ if and only if $x=x^{\prime}$.

  3. (iii)

The map $(x,y)\mapsto(F(x,y),y)$ is $C_{Lip}$-bi-Lipschitz on the set $g(E)$.
In particular, for all $y\in\mathbb{R}^{m}$, the restriction
$F|_{(\mathbb{R}^{n}\times\\{y\\})\cap g(E)}$ is $C_{Lip}$-bi-Lipschitz.

If $E\subseteq Q_{0}$ is a set and there exists a mapping
$g\colon\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n+m}$ satisfying (i)-(iii) for
$E$, then we call $E$ a Hard Sard set for $f$.

We think of $g$ as a globally bi-Lipshitz change of coordinates that
“straightens out” the fibers of $f|_{E}$. Since the linear projection mapping
$\pi(x,y)=x$ on $Q_{0}$ satisfies all the properties requested for the map
$F=f\circ g^{-1}$ on $g(E)$ in Definition 1.11, we interpret Definition 1.11
as saying that $f$ “looks like a projection” when restricted to $E$, up to
globally a bi-Lipschitz change of coordinates $g$. See Figure 1.

Figure 1. A schematic diagram of a Hard Sard set in the case $n=m=1$. The map
$g$ straightens out the (blue) fibers of $f$ and the (red) transverse curves
on which $f$ is bi-Lipschitz.

The motivation for both of the above definitions was Theorem I of Azzam and
the second author in [1]. That result was improved in [5] to the following
theorem.

######  Theorem 1.12 (Theorem A of [5]).

Let $Q_{0}$ be the unit cube in $\mathbb{R}^{n+m}$ and let $f\colon
Q_{0}\rightarrow X$ be a $1$-Lipschitz map into a metric space $X$ with
${\mathcal{H}}^{n}(f(Q_{0}))\leq 1.$ Given any $\gamma>0$, we can write
$Q_{0}=E_{1}\cup\dots\cup E_{M}\cup G,$ where $E_{i}$ are Hard Sard sets for
$f$ and ${\mathcal{H}}^{n,m}_{\infty}(f,G)<\gamma.$ The constant $M$ and the
constants $C_{Lip}$ associated to the Hard Sard pairs $(E_{i},g_{i})$ depend
only on $n$, $m$, and $\gamma$.

This theorem and [1, Theorem I] can be viewed as “quantitative implicit
function theorems” for Lipschitz maps into metric spaces. Compare also with
the qualitative theorem of [10].

Thus, in combination with Theorem 1.12 (or Theorem I of [1]), our Theorem A
immediately implies the following.

######  Corollary 1.13.

Let $Q_{0}$ be the unit cube in $\mathbb{R}^{2+m}$ and let $f\colon
Q_{0}\rightarrow\ell^{\infty}$ be a $1$-Lipschitz map with
${\mathcal{H}}^{n}(f(Q_{0}))\leq 1.$

If

$\sup_{x\in Q_{0}}\|g(x)-f(x)\|_{\infty}\geq\epsilon$

for all mappings $g\colon Q_{0}\rightarrow\ell^{\infty}$ that factor through
trees, then $f$ admits a Hard Sard set $E$ with $\mathcal{H}^{n+m}(E)>\gamma$,
where $\gamma>0$ depends only on $m$ and $\epsilon$.

###### Proof.

By Theorem A, we have $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})$ bounded below
depending only on $\epsilon$. By Theorem 1.12, $f$ then admits a Hard Sard set
$E$ with $\mathcal{H}^{n,m}_{\infty}(f,E)$ (and the associated constants)
bounded below. The $\mathcal{H}^{n+m}$-measure of $E$ is then comparable to
$\mathcal{H}^{n,m}_{\infty}(f,E)$ by [5, Lemma 3.5]. ∎

###  1.2. Additional results on mapping content: lower bounds and continuity

Theorem A will follow from Theorem 1.8 and a new lower bound (Theorem C) and
continuity statement (Theorem D) for mapping content, which apply for general
$n,m\in\mathbb{N}$. For the remainder of the document, we will fix $n,m$ non-
negative integers. We will write $Q_{0}=[0,1]^{n+m}$ and points of
$\mathbb{R}^{n+m}$ as $(x,y)$ where $x\in\mathbb{R}^{n}$ and
$y\in\mathbb{R}^{m}$. Later on, we will specialize to the case $n=2$ to prove
Theorem A.

Our theorem on lower-bounding mapping content is an extension of a result of
Kinneberg [14, Corollary 4.2]. Kinneberg’s result gives a simple way to lower
bound the Hausdorff content $\mathcal{H}^{d}_{\infty}(f([0,1]^{d}))$ of the
image of a continuous map defined on the unit cube, and we extend his methods
to apply to the mapping content $\mathcal{H}^{n,m}_{\infty}$.

Continuing to set $Q_{0}=[0,1]^{n+m}$, we let

(1.1)
$(F_{1},F^{\prime}_{1}),(F_{2},F^{\prime}_{2}),\dots,(F_{n+m},F^{\prime}_{n+m})$

denote all the pairs of opposite faces of $\partial Q_{0}$. Thus,

$F_{k}=\\{(t_{1},t_{2},\dots,t_{k-1},0,t_{k+1},\dots,t_{n+m}):t_{i}\in[0,1]\\}$

and

$F^{\prime}_{k}=\\{(t_{1},t_{2},\dots,t_{k-1},1,t_{k+1},\dots,t_{n+m}):t_{i}\in[0,1]\\}.$

######  Theorem C.

Let $f\colon Q_{0}\rightarrow Y$ be a continuous map into a metric space. Then

(1.2) $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})\geq\prod_{k=1}^{n}{\rm
dist}_{Y}(f(F_{k}),f(F^{\prime}_{k})).$

Moreover, the same lower bound holds for the quantity
$\hat{\mathcal{H}}^{n,m}_{\infty}$ defined in (2.1):

(1.3) $\hat{\mathcal{H}}^{n,m}_{\infty}(f,Q_{0})\geq\prod_{k=1}^{n}{\rm
dist}_{Y}(f(F_{k}),f(F^{\prime}_{k})).$

The quantity $\hat{\mathcal{H}}^{n,m}_{\infty}$ is a variant of
$\mathcal{H}^{n,m}_{\infty}$ introduced in [5, Section 1.4.2] that is more
convenient in some circumstances; see (2.1) below.

######  Remark 1.14.

In the case $m=0$, we have
$\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=\mathcal{H}^{n}_{\infty}(f(Q_{0}))$, so
this agrees with [14, Corollary 4.2]. Indeed, our proof follows the same path
as Kinneberg’s proof, including relying on his [14, Proposition 4.1].

######  Remark 1.15.

Of course, the fact that we use only the first $n$ pairs of sides in Theorem C
is irrelevant, as one can always reduce to this case.

Theorem C is of interest in its own right, providing a simple lower bound for
mapping content which then implies the existence of Hard Sard sets as
described above. In this paper, it will be used to prove a continuity
statement for mapping content.

For this, we will use the following notion of distance for Lipschitz maps into
arbitrary metric spaces, which is essentially an ad hoc notion of Gromov-
Hausdorff convergence which suffices for our purposes.

######  Definition 1.16.

Let $f\colon Q_{0}\rightarrow Y$ and $g\colon Q_{0}\rightarrow Z$ be
continuous functions onto metric spaces $Y$ and $Z$. For $\epsilon>0$, we will
say that ${\rm dist}(f,g)<\epsilon$ if there are isometric embeddings

$\iota_{Y}:Y\rightarrow\ell^{\infty}\text{ and
}\iota_{Z}:Z\rightarrow\ell^{\infty}$

such that

$\sup_{x\in Q_{0}}\|\iota_{Y}(f(x))-\iota_{Z}(g(x))\|_{\infty}<\epsilon.$

######  Remark 1.17.

To be concrete and avoid technicalities, we assume in the definition of ${\rm
dist}$ that $Y$ and $Z$ are the full images of $f$ and $g$, respectively,
hence the word “onto” in the definition. Of course, one can always reduce to
this case.

Our continuity statement for mapping content is then the following result.

######  Theorem D.

Let $Q_{0}=[0,1]^{n+m}$. Let $f_{i}\colon Q_{0}\rightarrow Y_{i}$ be a
sequence of $1$-Lipschitz maps onto metric spaces $Y_{i}$, and $f\colon
Q_{0}\rightarrow Y$ another $1$-Lipschitz map onto a metric space $Y$. Assume
that ${\rm dist}(f_{i},f)\rightarrow 0$ as $n\rightarrow\infty$. Then

$\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=0\text{ if and only if
}\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})\rightarrow 0\text{ as
}i\rightarrow\infty.$

Theorem D will then combine with Theorem 1.8 to prove Theorem A by a
compactness argument.

### Acknowledgments

The authors thank Behnam Esmayli and Piotr Hajłasz for comments, and for
sharing an early draft of [6].

##  2\. Preliminaries

###  2.1. Dyadic cubes

We write $Q_{0}$ for the unit cube in $\mathbb{R}^{d}$, with $d$ generally
understood from context, i.e.,

$Q_{0}=[0,1]^{d}.$

We write $\Delta$ for the collection of all dyadic cubes $Q\subseteq Q_{0}$,
and $\Delta_{k}$ for the collection of those dyadic cubes with side length
$2^{-k}$.

If $Q\in\Delta$, we write $\textnormal{side}(Q)$ for the side-length of $Q$.
Thus, $\textnormal{side}(Q)=2^{-k}$ if and only if $Q\in\Delta_{k}$.

If $Q\in\Delta$ and $C>0$, we write $CQ$ for a cube with the same center but
$C$ times the side length. In particular, if $C$ is an odd positive integer,
then $CQ$ is a union of $C^{d}$ distinct cubes of the same side length as $Q$.

Lastly, we call a collection of cubes “almost-disjoint” if they have disjoint
interiors. Such collections arise in the definition of
$\mathcal{H}^{n,m}_{\infty}$.

###  2.2. Metric derivatives

Let $X$ be a metric space and $f\colon\mathbb{R}^{d}\rightarrow X$ a
$1$-Lipschitz function. We will use results from [5], which rely on results
and notation from [2], which were in turn inspired by the idea of metric
differentiability in [15].

For a cube $Q\subseteq\mathbb{R}^{d}$ let

$\textnormal{md}_{f}(Q):=\frac{1}{\textnormal{side}(Q)}\inf_{\|\cdot\|}\sup_{x,y\in
Q}\left|d(f(x),f(y))-\|x-y\|\right|,$

where the infimum is taken over all seminorms $\|\cdot\|$ on $\mathbb{R}^{d}$.
If the function $f$ is understood, we will simply write $\textnormal{md}(Q)$.
The quantity $\textnormal{md}_{f}(Q)$ measures how well the pullback of the
distance in $X$ under $f$ can be approximated by a seminorm in $Q$. For metric
space valued functions, it serves as a replacement for measuring “deviation
from linearity”. This quantity is appealing because of the result of [2],
which is a quantitative differentiation result for Lipschitz mappings into
metric spaces:

######  Theorem 2.1 ([2], Theorem 1.1).

Let $X$ be a metric space and $f\colon\mathbb{R}^{d}\rightarrow X$ a
$1$-Lipschitz function. Let $\epsilon>0$ and $C_{0}>0$. Then

$\sum\\{|Q|:Q\in\Delta,\textnormal{md}_{f}(C_{0}Q)>\epsilon\\}\leq
C_{\epsilon,d}.$

The constant $C_{\epsilon,d}$ depends only on $\epsilon$, $C_{0}$, and $d$ but
not on the space $X$ or the function $f$.

Here $|Q|$ refers to the $d$-dimensional Lebesgue measure of $Q$. While we do
not use Theorem 2.1 directly below, it is behind the proof of Lemma 5.1 from
[5].

For a cube $Q$, we let $\|\cdot\|_{Q}$ denote a seminorm that gives rise to
the infimum in the definition above of md (repressing $f$ in the notation).
The existence of a minimizer follows from the observation that the norm is
determined by its unit ball, and such sets are compact under the Hausdorff
metric.

###  2.3. Content, measure, and variations

We use $|E|$ to denote the Lebesgue measure of a subset $E$ of some Euclidean
space, with the dimension understood from context.

For a subset $A$ of a metric space $X$, the $k$-dimensional Hausdorff content
$\mathcal{H}^{k}_{\infty}(A)$ is a well-used notion (see, e.g., [11, Chapter
8]) and is defined as follows:

$\mathcal{H}^{k}_{\infty}(A)=\inf\sum_{U\in\mathcal{U}}{\rm diam}(U)^{k},$

where the infimum is taken over all open covers $\mathcal{U}$ of $A$. (Small
modifications of the definition, e.g. to use covers by balls or arbitrary
sets, yield comparable quantities.) The quantity $\mathcal{H}^{k}_{\infty}$ is
countably sub-additive, but not in general a measure.

The notion of mapping content $\mathcal{H}^{n,m}_{\infty}$ was defined above
in Definition 1.4. As discussed in [1, 5], $\mathcal{H}^{n,m}_{\infty}$ serves
in some sense as a “coarse” substitute for the $L^{1}$-norm of the
$n$-dimensional Jacobian of $f$.

In [5, Section 1.4.2], a variation of mapping content was defined that uses
arbitrary sets in the cover, rather than dyadic cubes.

(2.1)
$\hat{\mathcal{H}}^{n,m}_{\infty}(f,A)=\inf\sum_{i}\mathcal{H}^{n}_{\infty}(f(S_{i})){\rm
diam}(S_{i})^{m},$

where the infimum is over all countable covers $\\{S_{i}\\}$ of $A$ by
arbitrary subsets of $Q_{0}$. It is immediately clear that

(2.2)
$\hat{\mathcal{H}}^{n,m}_{\infty}(f,A)\lesssim_{n,m}\mathcal{H}^{n,m}_{\infty}(f,A),$

As discussed in [5], we do not know if the reverse inequality holds in (2.2).
We do however know that if $\hat{\mathcal{H}}^{n,m}_{\infty}(f,A)=0$ then
$\mathcal{H}^{n,m}_{\infty}(f,A)=0$. Indeed we have the following more
quantitative statement.

######  Theorem 2.2 (Corollary E of [5]).

For each $\delta>0$, there is a $\delta^{\prime}>0$ with the following
property:

If $f:Q_{0}\rightarrow X$ is a $1$-Lipschitz mapping into a metric space, and
$A\subseteq Q_{0}$ has

$\mathcal{H}^{n,m}_{\infty}(f,A)\geq\delta,$

then

$\hat{\mathcal{H}}^{n,m}_{\infty}(f,A)\geq\delta^{\prime}.$

The number $\delta^{\prime}$ depends only on $\delta$, $n$, and $m$.

##  3\. Facts about seminorms

Here we collect some lemmas about seminorms that we need, the main goal being
Lemma 3.2 below.

######  Lemma 3.1.

Let $\|\cdot\|$ be a seminorm on $\mathbb{R}^{n+m}$. Then there is a linear
map $A\colon\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n+m}$ such that
$|Av|\approx\|v\|$ for all $v\in\mathbb{R}^{n+m}$. The implied constant
depends only on $n$, $m$.

###### Proof.

Let

$V=(ker(\|\cdot\|))^{\bot}=\left(\\{v:\|v\|=0\\}\right)^{\bot},$

a subspace of $\mathbb{R}^{n+m}$ with some dimension $k\in\\{0,\dots,n+m\\}$.

Let $A_{0}\colon\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n+m}$ be the orthogonal
projection onto $V$. Notice that $\|A_{0}w\|=\|w\|$ for all
$w\in\mathbb{R}^{n+m}$, and that $\|\cdot\|$ is a a norm when restricted to
$V$.

Let $K\subset V$ be the unit ball of $\|\cdot\|$. A theorem of F. John says
(see e.g. [3]) that there is an ellipsoid $E\subset K$ and such that $K\subset
kE$, where $k$ is the dimension of $V$. Let $T$ be the linear map taking $E$
to the unit ball of $\mathbb{R}^{k}$, viewed as embedded in
$\mathbb{R}^{n+m}$. Then for any unit vector $v\in V$ we have $1\leq|T(v)|\leq
k$, and thus, setting $A=T\circ A_{0}$ we have for all $w\in\mathbb{R}^{n+m}$
that $\|w\|=\|A_{0}w\|\leq|Aw|\leq k\|A_{0}w\|=k\|w\|$. Since $k\leq n+m$ this
completes the proof. ∎

The following fact about seminorms is now a consequence of Lemma 3.1 and the
singular value decomposition of matrices.

######  Lemma 3.2.

Let $\|\cdot\|$ be a seminorm on $\mathbb{R}^{n+m}$. Suppose that there is an
$n$-dimensional subspace $P\subseteq\mathbb{R}^{n+m}$ and a constant
$\delta>0$ such that

$\|v\|\geq\delta|v|\text{ for all }v\in P.$

Then there is an orthonormal basis $\\{v_{1},\dots,v_{n+m}\\}$ of
$\mathbb{R}^{n+m}$ such that

$\left\|\sum_{i=1}^{n+m}a_{i}v_{i}\right\|\gtrsim\delta\sum_{i=1}^{n}|a_{i}|$

for all choices of $a_{1},\dots,a_{n+m}$ in $\mathbb{R}$. The implied constant
depends only on $n$, $m$.

###### Proof.

Let $A\colon\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n+m}$ be as in Lemma 3.1.

Let $A=USV^{*}$ denote the singular value decomposition of $A$, with singular
values $\sigma_{1}\geq\dots\geq\sigma_{n+m}$. Thus, $U$ and $V^{*}$ are
orthogonal $(n+m)\times(n+m)$ matrices and $S$ is a diagonal
$(n+m)\times(n+m)$ matrix with the singular values along the diagonal. The
max-min characterization of singular values says that the $n$th largest
singular value, $\sigma_{n}$, can be computed by

$\sigma_{n}=\max_{\dim(T)=n}\min_{0\neq x\in
T}\frac{|Ax|}{|x|}\gtrsim\max_{\dim(T)=n}\min_{0\neq x\in
T}\frac{\|x\|}{|x|},$

where the maximum is over all subspaces $T$ of dimension $n$ in
$\mathbb{R}^{n+m}$.

Taking $T=P$, we obtain

$\sigma_{n}\gtrsim\delta>0,$

with implied constant depending only on $n,m$.

For $i=1,\dots,n+m$, let $v_{i}=(V^{*})^{-1}(e_{i})$, where $\\{e_{i}\\}$ is
the standard basis of $\mathbb{R}^{n+m}$. Then for any choice of
$a_{1},\dots,a_{n+m}$ in $\mathbb{R}$, we have

$\displaystyle\left\|\sum_{i=1}^{n+m}a_{i}v_{i}\right\|$
$\displaystyle\gtrsim\left|A\left(\sum_{i=1}^{n+m}a_{i}v_{i}\right)\right|$
$\displaystyle=\left|\sum_{i=1}^{n+m}a_{i}Se_{i}\right|$
$\displaystyle\gtrsim\delta\sum_{i=1}^{n}|a_{i}|,$

with implied constants depending only on $n$ and $m$. ∎

##  4\. Lower bounds for mapping content

This section is devoted to the proof of Theorem C.

###  4.1. A result of Kinneberg

We describe here the set-up of Kinneberg [14, Section 4] which we require.

Let $g\colon[0,1]^{d}\rightarrow X$ be a continuous map into a metric space.
Let $\\{U_{i}\\}_{i\in I}$ be an open cover of $g([0,1]^{d})$. A “chain” of
open sets from this collection is a finite sequence
$U_{i_{1}},U_{i_{2}},\dots,U_{i_{T}}$ such that consecutive sets intersect. We
say that the chain _connects_ two subsets $E,F\subseteq g([0,1]^{d})$ if $E$
intersects $U_{i_{1}}$ and $F$ intersects $U_{i_{T}}$.

For each $1\leq k\leq d$, let

$w_{k}\colon I\rightarrow[0,\infty)$

be a “weight function” defined on the sets of the cover.

For subsets $E,F\subseteq g([0,1]^{d})$, their “combinatorial distance” with
respect to the above choices is defined as

${\rm dist}_{w_{k}}(E,F)=\inf\sum_{t=1}^{T}w_{k}(i_{t}),$

where the infimum is taken over all chains $U_{i_{1}},\dots,U_{i_{T}}$
connecting $E$ to $F$.

Proposition 4.1 of [14] is then the following:

######  Proposition 4.1 (Kinneberg [14]).

For any continuous $g\colon[0,1]^{d}\rightarrow X$ and any open cover
$\\{U_{i}\\}_{I}$ and weight functions $w_{k}$ as defined above, we have

(4.1) $\sum_{i\in
I}\left(\prod_{k=1}^{d}w_{k}(i)\right)\geq\prod_{k=1}^{d}{\rm
dist}_{w_{k}}(g(F_{k}),g(F^{\prime}_{k})).$

Recall that $\\{F_{k},F^{\prime}_{k}\\}$ denote opposite faces of the boundary
of the unit cube, as defined in (1.1).

###  4.2. The proof of Theorem C

###### Proof of Theorem C.

We will prove the second statement (1.3) of Theorem C, and then indicate the
minor modifications needed to obtain the first statement (1.2).

Let $f$ be as in the statement of the theorem. Let $\\{S_{i}\\}_{i\in I}$ be
an arbitrary cover of $Q_{0}$ by sets $S_{i}$. For each $i\in I$, let
$\\{U^{i}_{j}\\}_{j\in J_{i}}$ be an arbitrary open cover of $f(S_{i})$. Fix
$\eta>0$ arbitrary.

For each $i\in I$ and $j\in J_{i}$, let

$V^{i}_{j}=U^{i}_{j}\times N_{\eta}(\text{Proj}_{y}(S_{i}))\subseteq
Y\times[0,1]^{m}.$

Here $\text{Proj}_{y}$ denotes the map $(x,y)\mapsto y$ from $[0,1]^{n+m}$ to
$[0,1]^{m}$, and $N_{\eta}(\cdot)$ denotes the open $\eta$-neighborhood of a
set in $[0,1]^{m}$. The purpose of taking the neighborhood $N_{\eta}$ in
$[0,1]^{m}$ is just to ensure that each set $V^{i}_{j}$ is open.

We now define weights $w_{k}(i,j)\in[0,\infty)$ for $1\leq k\leq n+m$, $i\in
I$, and $j\in J_{i}$.

$w_{k}(i,j)=\begin{cases}{\rm diam}(U^{i}_{j})&1\leq k\leq n\\\ {\rm
diam}(S_{i})+2\eta&n+1\leq k\leq n+m\end{cases}$

We will apply Theorem 4.1 to the function $h\colon Q_{0}\rightarrow
Y\times[0,1]^{m}$ defined by $h(x,y)=(f(x,y),y).$ We equip $Y\times[0,1]^{m}$
with the metric $d_{Y\times[0,1]^{m}}$ defined as
$d_{Y\times[0,1]^{m}}((p,t),(q,s))=\max\\{d_{Y}(p,q),|t-s|\\}$. In this
application, the open cover of Theorem 4.1 will be $\\{V^{i}_{j}\\}$, and the
weights $w_{k}(i,j)$ will be as defined above. The left-hand side of (4.1) is
then

(4.2) $\sum_{i\in I}\sum_{j\in J_{I}}\prod_{k=1}^{n+m}w_{k}(i,j)=\sum_{i\in
I}\left(\sum_{j\in J_{i}}{\rm diam}(U^{i}_{j})^{n}\right)({\rm
diam}(S_{i})+2\eta)^{m}.$

After taking the infimum over all choices of $\\{S_{i}\\}$, $\\{U^{i}_{j}\\}$,
and $\eta$, this yields exactly $\hat{\mathcal{H}}^{n,m}_{\infty}(f,Q_{0})$.

We now lower bound the right-hand side of (4.1). First, for $1\leq k\leq n$,
we have

$\displaystyle{\rm dist}_{w_{k}}(h(F_{k}),h(F^{\prime}_{k}))$
$\displaystyle=\inf\\{\sum_{t}w_{k}(V^{i_{t}}_{j_{t}}):\\{V^{i_{t}}_{j_{t}}\\}\text{
connects }h(F_{k}),h(F^{\prime}_{k})\\}$ $\displaystyle=\inf\\{\sum_{t}{\rm
diam}(U^{i}_{j}):\\{V^{i_{t}}_{j_{t}}\\}\text{ connects
}h(F_{k}),h(F^{\prime}_{k})\\}$ $\displaystyle\geq\inf\\{\sum_{t}{\rm
diam}(U^{i}_{j}):\\{U^{i_{t}}_{j_{t}}\\}\text{ connects
}f(F_{k}),f(F^{\prime}_{k})\\}$ $\displaystyle\geq{\rm
dist}_{Y}(f(F_{k}),f(F^{\prime}_{k})).$

Next, consider $n+1\leq k\leq n+m$. Let

$A=\\{(y_{1},\dots,y_{k-n-1},0,y_{k-n+1},\dots,y_{m}\\}\subseteq[0,1]^{m}$

and

$B=\\{(y_{1},\dots,y_{k-n-1},1,y_{k-n+1},\dots,y_{m}\\}\subseteq[0,1]^{m}.$

We then have

$\displaystyle{\rm dist}_{w_{k}}(h(F_{k}),h(F^{\prime}_{k}))$
$\displaystyle=\inf\\{\sum_{t}w_{k}(V^{i_{t}}_{j_{t}}):\\{V^{i_{t}}_{j_{t}}\\}\text{
connects }h(F_{k}),h(F^{\prime}_{k})\\}$ $\displaystyle=\inf\\{\sum_{t}({\rm
diam}(S_{i})+2\eta):\\{V^{i_{t}}_{j_{t}}\\}\text{ connects
}h(F_{k}),h(F^{\prime}_{k})\\}$ $\displaystyle\geq\inf\\{\sum_{t}({\rm
diam}(S_{i})+2\eta):\\{N_{\eta}(\text{Proj}_{k}(S_{i}))\\}\text{ connects
}A,B\\}$ $\displaystyle\geq 1$

It follows that the right-hand side of (4.1) is bounded below by

$\prod_{k=1}^{n}{\rm dist}_{Y}(f(F_{k}),f(F^{\prime}_{k})),$

independently of the choice of sets $\\{S_{i}\\}$, open sets
$\\{U^{i}_{j}\\}$, and $\eta>0$. Taking the infimum over all these choices, we
obtain (1.3).

The proof of (1.2) is the same, with the following minor modifications: The
sets $S_{i}$ are replaced by almost-disjoint dyadic cubes $Q_{i}$. The role of
the open sets $U^{j}_{i}$ and parameter $\eta>0$ remain the same. The weight
$w_{k}(i,j)$ is the same as before if $1\leq k\leq n$ and changes to
$\textnormal{side}(Q_{i})+2\eta$ if $n+1\leq k\leq n+m$.

With these modifications, taking the infimum over all collections
$\\{Q_{i}\\}$, $\\{U^{i}_{j}\\}$, and parameters $\eta>0$ in the analog of
(4.2) yields $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})$, and the calculations with
the modified weights $w_{k}(i,j)$ yield the same lower bound. ∎

Theorem C will enter the proof of Theorem D through the following corollary.

######  Corollary 4.2.

Let $f\colon Q_{0}\rightarrow Y$ be $1$-Lipschitz and $c,\eta>0$. Let
$Q\subseteq Q_{0}$ be a cube, and let $\|\cdot\|$ be a seminorm such that

$\left|d(f(x),f(y))-\|x-y\|\right|<\eta\textnormal{side}(3Q)\text{ for all
}x,y\in 3Q.$

Suppose that there is an $n$-plane $P\subseteq\mathbb{R}^{n+m}$ such that
$\|v\|\geq c|v|$ for all $v\in P$.

If $\eta$ is sufficiently small, depending only on $n,m,c$, then

$\mathcal{H}^{n,m}_{\infty}(f,Q)\gtrsim|Q|,$

with implied constant depending only on $n$, $m$, and $c$.

###### Proof.

Let $v_{1},\dots,v_{n+m}$ be an orthonormal basis of $\mathbb{R}^{n+m}$
“adapted” to $\|\cdot\|$, as provided by Lemma 3.2. Let $Q^{\prime}\subseteq
Q$ be a rotated cube with
$\textnormal{side}(Q^{\prime})\approx_{n,m}\textnormal{side}(Q)$ oriented
along the axes $v_{1},\dots,v_{n+m}$. Let
$\\{(F_{k},F^{\prime}_{k})\\}_{k=1}^{n+m}$ denote the pairs of opposite faces
of $Q^{\prime}$.

Let $1\leq k\leq n$ and let $x\in F_{k},x^{\prime}\in F^{\prime}_{k}$. Let
$v=x^{\prime}-x$. Writing $v=\sum_{i=1}^{n+m}a_{i}v_{i}$ in the new basis, we
have $a_{k}\gtrsim\textnormal{side}(Q)$. By choice of seminorm,

$d(f(x),f(x^{\prime}))\geq\|v\|-3\eta\textnormal{side}(Q).$

By Lemma 3.2, $\|v\|\gtrsim c|a_{k}|\gtrsim c\textnormal{side}(Q)$, with
implied constant depending only on $n,m$. Thus, if $\eta$ is chosen
sufficiently small depending on $n$, $m$, and $c$, then
$d(f(x),f(x^{\prime}))$ is bounded below away from zero by
$a\textnormal{side}(Q)$, for some constant $a$ depending only on $n$, $m$, and
$c$.

This proves that

${\rm dist}(f(F_{k}),f(F^{\prime}_{k}))\geq a\textnormal{side}(Q)\text{ for
}1\leq k\leq n,$

and therefore by statement (1.3) of Theorem C (rescaled and rotated to apply
to $Q^{\prime}$) that

$\hat{\mathcal{H}}^{n,m}_{\infty}(f,Q^{\prime})\gtrsim|Q^{\prime}|,$

with implied constant depending on $n$, $m$, and $c$.

Therefore, we have

$\mathcal{H}^{n,m}_{\infty}(f,Q_{0})\gtrsim_{n,m}\hat{\mathcal{H}}^{n,m}_{\infty}(f,Q_{0})\geq\hat{\mathcal{H}}^{n,m}_{\infty}(f,Q^{\prime})\gtrsim_{n,m,c}|Q^{\prime}|\gtrsim_{n,m}|Q|.$

∎

##  5\. Zero mapping content passes to the limit

This section is devoted to the proof of Theorem D. We begin with some lemmas.
The first is Lemma 7.3 of [5] (stated in a slightly different way).

######  Lemma 5.1.

Given $\delta>0$, there is a constant $c>0$, depending only on $n$, $m$, and
$\delta$, with the following property.

Let $Q_{0}=[0,1]^{n+m}$ and let $f\colon Q_{0}\rightarrow Y$ be $1$-Lipschitz.
Suppose that $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=\delta>0$ and $\eta>0$. Then
there is a cube $Q\subseteq Q_{0}$ such that

$\textnormal{md}_{f}(3Q)<\eta$

and an $n$-plane $P$ in $\mathbb{R}^{n+m}$ such that

(5.1) $\|v\|_{Q}\geq c|v|\text{ for all }v\in P.$

Moreover, the size of the cube $Q$ can be bounded below by a constant
depending only on $n$, $m$, $\delta$, and $\eta$.

######  Remark 5.2.

In fact, the proof in [5] shows that $P$ can be chosen to be a coordinate
plane, though we do not need this here.

###### Proof of Theorem D.

Let $\\{f_{i}\colon Q_{0}\rightarrow Y_{i}\\}$ and $f\colon Q_{0}\rightarrow
Y$ be as in the statement of Theorem D.

Suppose first that $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=\delta>0$; we will
show that $\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})\rightarrow 0$.

Let $c>0$, depending on $\delta,n,m$, be as in Lemma 5.1. Let $\eta$ be chosen
sufficiently small, depending on $n,m,c$, as required by Corollary 4.2.

By Lemma 5.1, we may find a cube $Q\subseteq Q_{0}$ such that

$\textnormal{md}_{f}(3Q)<\eta/10$

and such that there is an $n$-plane $P$ as in (5.1), with respect to a
seminorm $\|\cdot\|_{Q}$ on $\mathbb{R}^{n+m}$ such that

$|d(f(x),f(y))-\|x-y\|_{Q}|<\frac{\eta}{10}\textnormal{side}(3Q)\text{ for all
}x,y\in 3Q.$

Choose $i_{0}$ sufficiently large so that ${\rm
dist}(f_{i},f)<\frac{\eta}{10}\textnormal{side}(Q)$ for all $i\geq i_{0}$. The
same seminorm $\|\cdot\|_{Q}$ satisfies

$|d(f_{i}(x),f_{i}(y))-\|x-y\|_{Q}|<\frac{3\eta}{10}\textnormal{side}(3Q)\text{
for all }x,y\in 3Q\text{ and }i\geq i_{0}$

It follows from Corollary 4.2 that

$\mathcal{H}^{n,m}_{\infty}(f_{i},Q)\gtrsim_{n,m,\delta}|Q|$

for all $i$ sufficiently large, which contradicts the assumption that
$\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})\rightarrow 0$.

For the converse statement, we continue to assume that ${\rm
dist}(f_{i},f)\rightarrow 0$, but now we suppose that
$\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=0$ and that
$\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})$ does not tend to zero. In that case,
passing to a subsequence (which we rename $\\{f_{i}\\}$ to keep the notation
the same) yields $\delta>0$ such that
$\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})\geq\delta>0$. Exactly as before, we
set $c=c(\delta,n,m)>0$ as in Lemma 5.1 and $\eta=\eta(n,m,c)$ sufficiently
small as required by Corollary 4.2.

It follows that for each $i$, there is a dyadic cube $Q_{i}\subseteq Q_{0}$
and an $n$-plane $P_{i}$ as in (5.1), with respect to a seminorm
$\|\cdot\|_{i}$ on $\mathbb{R}^{n+m}$ such that

$|d(f_{i}(x),f_{i}(y))-\|x-y\|_{i}|<\frac{\eta}{10}\textnormal{side}(3Q_{i})\text{
for all }x,y\in 3Q_{i}.$

Moreover, Lemma 5.1 guarantees that $\textnormal{side}(Q)\geq s_{0}$, for some
$s_{0}$ depending only on $n,m,\delta,\eta$, and hence $n,m,\delta$.

Choose any $i$ sufficiently large so that ${\rm dist}(f,f_{i})<\eta
s_{0}/10\leq\eta\textnormal{side}(Q_{i})/10$. The same seminorm
$\|\cdot\|_{i}$ satisfies

$|d(f(x),f(y))-\|x-y\|_{i}|<\frac{3\eta}{10}\textnormal{side}(3Q)\text{ for
all }x,y\in 3Q_{i}\text{ and }i\geq i_{0}$

It therefore again follows from Corollary 4.2 that

$\mathcal{H}^{n,m}_{\infty}(f,Q_{0})\geq\mathcal{H}^{n,m}_{\infty}(f,Q_{i})\gtrsim_{n,m,\delta}|Q_{i}|>0$

contradicting the assumption that $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=0$.

∎

##  6\. Proofs of Theorem B and Theorem A

In this section, we prove the two approximation theorems, Theorems B and A.
The latter will be an immediate corollary of the former and Theorem 1.8.

We first need a Gromov-Hausdorff type compactness lemma.

######  Lemma 6.1.

Fix $d\in\mathbb{N}$. Let $f_{i}\colon[0,1]^{d}\rightarrow Y_{i}$ be a
sequence of $1$-Lipschitz mappings onto metric spaces $Y_{i}$. Then there is a
$1$-Lipschitz mapping $f\colon[0,1]^{d}\rightarrow Y$ onto a metric space and
a subsequence $\\{f_{i_{k}}\\}$ such that

${\rm dist}(f_{i_{k}},f)\rightarrow 0.$

###### Proof.

We will first use Gromov’s compactness theorem (see [4, Theorem 7.4.15]) to
argue that a subsequence of the spaces $Y_{i}$ converge in the Gromov-
Hausdorff topology. First of all, each $Y_{i}$ has

(6.1) ${\rm diam}(Y_{i})\leq{\rm diam}([0,1]^{d})=\sqrt{d}\text{ uniformly in
}i,$

as a $1$-Lipschitz image of the unit cube. Second of all, for every
$\epsilon>0$, let $N(X,\epsilon)$ be the maximal size of an
$\epsilon$-separated set in a metric space $X$. Observing that the pre-image
of an $\epsilon$-separated set under a $1$-Lipschitz map is again
$\epsilon$-separated, we have

(6.2) $N(Y_{i},\epsilon)\leq N([0,1]^{d},\epsilon)\lesssim\epsilon^{-d}\text{
uniformly in }i.$

The properties (6.1) and (6.2) of the sequence $\\{Y_{i}\\}$ imply that the
sequence contains a Gromov-Hausdorff convergent subsequence, by [4, Theorem
7.4.15]. For convenience, let us continue to label this subsequence
$\\{Y_{i}\\}$.

We may now find isometric embeddings $\iota_{i}\colon
Y_{i}\rightarrow\ell^{\infty}$ and a compact subset $Y\subseteq\ell^{\infty}$
such that

$d_{\text{Haus}}(\iota_{i}(Y_{i}),Y)\rightarrow 0,$

where $d_{\text{Haus}}$ denotes the Hausdorff distance in $\ell^{\infty}$.
(See [12, Proposition 2.8].)

By a standard diagonalization argument, we may now extract a subsequence
$\\{f_{i_{k}}\\}$ of our original sequence and a $1$-Lipschitz map $f\colon
Q_{0}\rightarrow Y$ such that

$\sup_{Q_{0}}\|\iota_{i_{k}}\circ f_{i_{k}}-f\|_{\infty}\rightarrow 0.$

It follows that

${\rm dist}(f_{i_{k}},f)\rightarrow 0.$

∎

We will also use the following simple fact.

######  Lemma 6.2.

Let $f\colon Q_{0}\rightarrow Y$ and $\phi\colon Y\rightarrow Z$ be
$1$-Lipschitz maps. Then

$\mathcal{H}^{n,m}_{\infty}(\phi\circ
f,Q_{0})\leq\mathcal{H}^{n,m}_{\infty}(f,Q_{0}).$

###### Proof.

This follows immediately from the definition of mapping content and the fact
that

$\mathcal{H}^{n}_{\infty}(\phi(f(Q)))\leq\mathcal{H}^{n}_{\infty}(f(Q))$

for every cube $Q$. ∎

###### Proof Theorem B.

We begin with the first statement in the theorem: small mapping content
implies close to factoring through a tree. Suppose that the first statement in
the theorem were false. There would then be an $\epsilon>0$ and a sequence
$f_{i}\colon Q_{0}\rightarrow\ell^{\infty}$ of $1$-Lipschitz maps such that

$\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})\rightarrow 0$

but

(6.3) $\sup_{Q_{0}}\|f_{i}-g\|_{\infty}\geq\epsilon$

for all $i$ and all $1$-Lipschitz maps $g\colon Q_{0}\rightarrow\ell^{\infty}$
with $\mathcal{H}^{n,m}_{\infty}(g,Q_{0})=0$.

By Lemma 6.1, there is a $1$-Lipschitz map $f$ from $Q_{0}$ onto a metric
space $Y$ such that

${\rm dist}(f_{i},f)\rightarrow 0.$

By Theorem D, we have $\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=0$.

Choose $i$ large such that ${\rm dist}(f_{i},f)<\epsilon$, and write
$Y_{i}=f_{i}(Q_{0})$. There are isometric embeddings

$\iota\colon Y_{i}\rightarrow\ell^{\infty}$

and

$\iota^{\prime}\colon Y\rightarrow\ell^{\infty}$

such that

$\sup_{Q_{0}}\|\iota\circ f_{i}-\iota^{\prime}\circ f\|_{\infty}<\epsilon.$

Let $j\colon\iota(Y_{i})\rightarrow Y_{i}\subseteq\ell^{\infty}$ be the
inverse of $\iota$. We may extend $j$ to a $1$-Lipschitz map from
$\ell^{\infty}$ to itself. Let $g=j\circ\iota^{\prime}\circ f$. Note that,
like $f$, $g$ is a $1$-Lipschitz map with
$\mathcal{H}^{n,m}_{\infty}(g,Q_{0})=0$, by Lemma 6.2.

In addition,

$\displaystyle\sup_{Q_{0}}\|f_{i}-g\|_{\infty}$
$\displaystyle=\sup_{Q_{0}}\|j\circ\iota\circ f_{i}-j\circ\iota^{\prime}\circ
f\|_{\infty}$ $\displaystyle\leq\sup_{Q_{0}}\|\iota\circ
f_{i}-\iota^{\prime}\circ f\|_{\infty}$ $\displaystyle<\epsilon$

This contradicts (6.3) and so completes the proof of the first half of Theorem
B.

We now consider the second, converse statement. Suppose that the second
statement in the theorem were false. There would then be an $\epsilon>0$ and
two sequences $f_{i},g_{i}\colon Q_{0}\rightarrow\ell^{\infty}$ of
$1$-Lipschitz maps such that

  * •

$\sup_{Q_{0}}\|f_{i}-g_{i}\|_{\infty}\rightarrow 0$,

  * •

$\mathcal{H}^{n,m}(g_{i},Q_{0})=0$ for each $i$, and

  * •

$\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})\geq\epsilon$ for each $i$ and some
fixed $\epsilon>0$.

By passing to a subsequence, we may again assume that there is a $1$-Lipschitz
map $f\colon Q_{0}\rightarrow Y$ such that ${\rm dist}(f_{i},f)\rightarrow 0$.

There are isometric embeddings $\iota_{i}\colon
f_{i}(Q_{0})\rightarrow\ell^{\infty}$, $\iota\colon
f(Q_{0})\rightarrow\ell^{\infty}$ such that

$\sup_{Q_{0}}\|\iota_{i}\circ f_{i}-\iota\circ f\|.$

As before, we may extend $\iota_{i}$ and $\iota$ to be $1$-Lipschitz maps
defined on all of $\ell_{\infty}$. Let $j$ be the inverse of
$\iota|_{f(Q_{0})}$, extended to be a $1$-Lipschitz map on $\ell_{\infty}$.

It follows that

$\displaystyle\sup_{Q_{0}}\|f-j\circ\iota_{i}\circ g_{i}\|_{\infty}$
$\displaystyle=\sup_{Q_{0}}\|j\circ\iota\circ f-j\circ\iota_{i}\circ
g_{i}\|_{\infty}$ $\displaystyle\leq\sup_{Q_{0}}\|\iota\circ f-\iota_{i}\circ
g_{i}\|_{\infty}$ $\displaystyle\leq\sup_{Q_{0}}\|\iota\circ f-\iota_{i}\circ
f_{i}\|_{\infty}+\sup_{Q_{0}}\|\iota_{i}\circ f_{i}-\iota_{i}\circ
g_{i}\|_{\infty}$ $\displaystyle\leq\sup_{Q_{0}}\|\iota\circ f-\iota_{i}\circ
f_{i}\|_{\infty}+\sup_{Q_{0}}\|f_{i}-g_{i}\|_{\infty}$
$\displaystyle\rightarrow 0$

Therefore, ${\rm dist}(f,j\circ\iota_{i}\circ g_{i})\rightarrow 0$. Since
$\mathcal{H}^{n,m}_{\infty}(g_{i},Q_{0})=0$ for each $i$, the same holds for
$\iota_{i}\circ g_{i}$ by Lemma 6.2. It follows from Theorem D that
$\mathcal{H}^{n,m}_{\infty}(f,Q_{0})=0$.

Again by Theorem D, we must therefore have
$\mathcal{H}^{n,m}_{\infty}(f_{i},Q_{0})\rightarrow 0$, but this contradicts
one of the defining properties of $f_{i}$. ∎

###### Proof of Theorem A.

The theorem now follows immediately from Theorem B and Theorem 1.8. ∎

## References

  * [1] Jonas Azzam and Raanan Schul.  Hard Sard: quantitative implicit function and extension theorems for Lipschitz maps.  Geom. Funct. Anal., 22(5):1062–1123, 2012.
  * [2] Jonas Azzam and Raanan Schul.  A quantitative metric differentiation theorem.  Proc. Amer. Math. Soc., 142(4):1351–1357, 2014.
  * [3] Keith Ball.  Ellipsoids of maximal volume in convex bodies.  Geometriae Dedicata, 41(2):241–250, 1992.
  * [4] Dmitri Burago, Yuri Burago, and Sergei Ivanov.  A course in metric geometry, volume 33.  American Mathematical Soc., 2001.
  * [5] Guy C. David and Raanan Schul.  Quantitative decompositions of Lipschitz mappings into metric spaces.  arXiv preprint arXiv:2002.10318, 2020.
  * [6] Behnam Esmayli and Piotr Hajłasz.  Lipschitz mappings, metric differentiability, and factorization through metric trees.  arXiv preprint arXiv:2106.15763, 2021.
  * [7] Behnam Esmayli and Piotr Hajłasz.  The coarea inequality.  arXiv preprint arXiv:2006.00419, 2020.
  * [8] H. Federer.  Geometric measure theory.  Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.
  * [9] P. Goldstein, P. Hajłasz, and P. Pankka.  Topologically nontrivial counterexamples to Sard’s theorem.  International Mathematics Research Notices, 08 2018.  rny179.
  * [10] P. Hajł asz and S. Zimmerman.  An implicit function theorem for Lipschitz mappings into metric space.  Indiana Univ. Math. J., 69(1):205–228, 2020.
  * [11] J. Heinonen.  Lectures on analysis on metric spaces.  Universitext. Springer-Verlag, New York, 2001.
  * [12] J. Heinonen.  Geometric embeddings of metric spaces.  Number 90. University of Jyväskylä, 2003.
  * [13] R. Kaufman.  A singular map of a cube onto a square.  J. Differential Geom., 14(4):593–594 (1981), 1979.
  * [14] Kyle Kinneberg.  Discrete length-volume inequalities and lower volume bounds in metric spaces.  Mathematische Zeitschrift, 282(3-4):747–768, 2016.
  * [15] B. Kirchheim.  Rectifiable metric spaces: local structure and regularity of the Hausdorff measure.  Proc. Amer. Math. Soc., 121(1):113–123, 1994.
  * [16] Stefan Wenger and Robert Young.  Lipschitz homotopy groups of the Heisenberg groups.  Geom. Funct. Anal., 24(1):387–402, 2014.