General methods of elliptic minimization

Abstract

We provide general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. Given a collection of competing “surfaces” that span a given “bounding set” in an ambient metric space, we produce one minimizing an elliptic area functional. The collection of competing surfaces is assumed to satisfy a set of geometrically-defined axioms. These axioms hold for collections defined using any combination of homological, cohomological or linking number spanning conditions. A variety of minimization problems can be solved, including sliding boundaries.

1 Introduction

Plateau’s problem asks whether there exists a surface of least area among those with a given boundary. It was named after the French physicist Joseph Plateau, who in the nineteenth century experimented with soap films and formulated laws that describe their structure. There is no single theorem or conjecture called Plateau’s problem; it is rather a general framework that has many precise formulations. Douglas [10] and Radó [27] independently solved the first such formulation of Plateau’s problem by finding an area minimizer among immersed parametrized disks with a prescribed boundary in $\mathbb {R}^n$. Three seminal papers appearing in 1960 [8, 17, 28] employed different definitions of “surface” and “boundary” and solved distinct versions of the problem. The techniques developed in these papers gave birth to the modern field of geometric measure theory.

We will briefly describe the problems solved in [17, 28], leaving proper definitions and details to the original sources. An m-rectifiable set equipped with a pointwise orientation and integer multiplicity can be integrated against differential forms. Such objects are called “rectifiable currents” and possess mass (m-dimensional Hausdorff measure weighted by the multiplicity) and a boundary operator (the dual to exterior derivative.) If the boundary of a rectifiable current is also rectifiable, it is called an “integral current.” Federer and Fleming [17] used these integral currents to define their competing surfaces and used mass to define “area”. On the other hand, Reifenberg [28] used compact sets for surfaces and Hausdorff measure to define area. There is no boundary operator defined for sets, so instead he turned to Čech homology to define a collection of competing “surfaces” with a given boundary. Roughly speaking, given a boundary set A and a set L of Čech cycles in A, a set $X\supset A$ is a competing surface if each cycle in L bounds in X.

There are advantages and disadvantages to using either sets or currents. Each approach has its applications and is suitable for different problems. Sets tend to be more difficult to work with than currents because of the lack of a boundary operator and the fact that unlike mass, Hausdorff measure is not lower-semicontinuous in any useful topology. A substantive difference between the two is that integral currents possess an orientation and sets do not. In practice, two currents with opposite orientations cancel when brought together, while sets do not. Sets are better models for physical soap films, since if two soap films touch, they merge rather than cancel.

Plateau’s problem requires minimization of an area functional $X \mapsto A (X)$ where X is a competing surface and $A$ refers to either mass or m-dimensional Hausdorff measure. The problem can be generalized to a heterogeneous problem by allowing the ambient density of $A$ to vary pointwise by a function f. In this case, one would minimize the functional $X \mapsto \int _X f(p) dA$. The heterogeneous problem itself is a special case of an anisotropic minimization problem in which the density function f can depend non-trivially on m-dimensional tangent directions. In this case, the functional would be $X\mapsto \int _X f(p,T_pX) dA$.

For example, consider the cost of building roads between several towns. If the land is flat and homogeneous but with a varying cost of acquisition, then the cost minimization problem is a heterogeneous but isotropic minimization problem. However, if the land is hilly with variable topography, then cost becomes an anisotropic problem.

Almgren [1] worked on the anisotropic minimization problem and defined a necessary ellipticity condition on the area functional. Roughly speaking, an anisotropic area functional is elliptic if an m-disk centered at any given point can be made small enough so that it very nearly minimizes the area functional among surfaces with the same $(m-1)$-sphere boundary. This ellipticity condition as defined in [1] is analogous to Morrey’s quasiconvexity used in parametric variational problems [24]. Federer [15] used a parametric variant of Almgren’s elliptic integrands to obtain an anisotropic version of [17] for integral currents.

In this paper, we establish the existence of an m-dimensional surface in an ambient metric space that minimizes an elliptic area functional for collections of sets satisfying axiomatic spanning conditions, including the collections considered in [1]. This solves a problem of geometric measure theory from the 1960’s (e.g., see [1, 2],) namely to provide an elliptic version of the “size minimization problem” as in Reifenberg [28]. “Size” in this context refers to Hausdorff measure.¹ Roughly speaking, given a bounding set A and a collection of m-rectifiable sets X that “span” A with respect to a geometrically-defined set of axioms Sect. 1.2.3, we prove there exists an element in the collection with minimum m-dimensional Hausdorff measure, weighted by an anisotropic density function. In Sect. 2.2 we describe a variety of topologically-defined collections that satisfy the axioms. Our methods build upon isotropic results in [22].

1.1 Recent history and current developments

In [21] we used linking numbers to specify spanning conditions: If M is an oriented $(n-2)$-dimensional connected submanifold of $\mathbb {R}^n$, we say a set $X\subset \mathbb {R}^n\, {{\varvec{spans}}}\, M$ if every circle embedded in the complement of M that has linking number one with M has non-trivial intersection with X. This definition, first introduced in [19, 20], can be extended to arbitrary codimension by replacing linking circles with spheres and to the case that M is not connected by specifying linking numbers with each component. We proved the following result, relying on [2] for regularity:

Theorem [21]

Let M be an oriented, compact $(n-2)$-dimensional submanifold of $\mathbb {R}^n$ and $\mathcal {S}$ the collection of compact sets spanning M. There exists an $X_0$ in $\mathcal {S}$ with smallest size. Any such $X_0$ contains a “core” $X_0^*\in \mathcal {S}$ with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of $(n-1)$-dimensional Hausdorff measure) a real analytic $(n-1)$-dimensional minimal submanifold.

De Lellis et al. [9] built upon our linking number spanning condition in [21]. They extracted the first axiomatic spanning conditions for codimension one surfaces using so-called “good classes” with “good comparison properties”. Roughly speaking, a collection of surfaces has “good comparison properties” if one cannot significantly reduce area by substituting local cone or cup competitors arising from Caccioppoli theory. (We refer to [9] for the full definitions). Their work provided a new proof to the main result of [21] and, simultaneously, a new approach to the “sliding boundary” problem posed by David [6] (see Sect. 2.6 for further discussion). De Philippis et al. [11] extended [9] to higher codimension, replacing links by simple closed curves with links by spheres.

Meanwhile, the authors independently extended [21] to higher codimensions in [22] using a spanning condition defined using cohomology. We also minimized Hausdorff measure weighted by an isotropic Hölder density function. By Alexander duality, taking geometric representatives for homology classes, this cohomological spanning condition is equivalent to the above linking condition, but in which the linking spheres are replaced with surfaces with possibly higher genus and conical singularities.

The isotropic density function of [22] is replaced by an anisotropic density in the current paper. At essentially the same time as this paper was announced, de Lellis, de Rosa and Ghiraldin posted their anisotropic paper [12] for codimension one. Their axiomatic spanning conditions built upon those presented in [9]. Our axioms in Sect. 1.2.3 require that a collection of sets be closed under Hausdorff limits and the action of diffeomorphisms keeping the bounding set A fixed. We note that all collections using homological, cohomological, or linking spanning conditions satisfy these conditions.

1.2 Advances in this paper

1.2.1 Ambient spaces

In this paper we permit the ambient space in which the minimization occurs to be a certain type of metric space that can be isometrically embedded in $\mathbb {R}^n$ as a Lipschitz retraction of some neighborhood of itself. See Definition 2.1. Examples include Riemannian manifolds with boundary and/or conical singularities (see Fig. 1)

Whenever one wishes to extend a particular result in geometric measure theory from Euclidean space to an ambient Riemannian manifold, one is faced with the choice of either working in charts, or embedding the manifold in Euclidean space and proving that the various constructions used can be deformed back onto the manifold. This second approach is usually much simpler, and yields further generalization to spaces more general than manifolds, namely Lipschitz neighborhood retracts. We have, for the most part, chosen this second approach. However, the full category of Lipschitz neighborhood retracts seems slightly out of reach. We make use of one construction in particular, namely Lemma 3.4, in which it is vital to assume slightly more about the ambient space, namely that the Lipschitz retraction can be “localized” (Definition 2.1). Nevertheless, this slightly restricted class of Lipschitz neighborhood retracts contains all the interesting examples we can think of, including finite simplicial complexes and Riemannian manifolds with boundary and certain singularities.

1.2.2 Bounding sets

In our approach to the size minimization problem, we begin with a fixed “bounding set” A. This is a compact set that all competitors X are assumed to contain.² The role that A plays in Plateau problems mimics that of a boundary condition in PDE’s, but we call A a “bounding set” since it might look nothing like the boundary of an m-dimensional set in some simple examples (see Fig. 3). We permit A to be any compact set, including the empty set and sets with dimension n (we shall minimize the elliptic functional over the sets $X{\setminus } A$).

Fig. 1

This pinched solid torus is an ambient space for anisotropic minimization problems. The bounding set A is the union of two (dashed) circles in the interior of the torus and the interior pinched (dashed) cylinder depicts a competitor spanning A

1.2.3 Geometrically defined axiomatic spanning conditions

The problem of finding axiomatic conditions on collections of sets sufficient to solve minimization problems was posed in [7]. The axioms presented in Sect. 2.2 assume that our collections of sets are closed under Lipschitz deformations and Hausdorff limits. These conditions are all met by the algebraic spanning conditions in Sect. 1.2.4. If the ambient space is a Riemannian manifold, then the Lipschitz deformations can be replaced by diffeomorphisms isotopic to the identity (see Definition 2.4).

1.2.4 Algebraic spanning conditions

Currents possess a boundary operator, and for minimization problems in which the “surfaces” are currents, this boundary operator can be used to specify a spanning condition. That is, a current S is said to span a current T if the boundary of S is T. However, there is no boundary operator for sets and it takes more work to specify spanning conditions for minimization problems involving sets. We are aware of two closely related types of algebraic spanning conditions that satisfy our axioms. The first is defined using Čech theory, either homological, cohomological or a combination of the two (see Definition 2.7). The second uses linking numbers as defined in Sect. 1.1, and is homotopical in nature (see Definition 2.8). The key property needed for both types is continuity under either weak or Hausdorff limits. That is, if $(X_i)_{i\in \mathbb {N}}$ is a minimizing sequence of surfaces satisfying a spanning condition, and $(X_i)_{i\in \mathbb {N}}$ converges to $X_0$, then $X_0$ should also satisfy the spanning condition. The Čech theoretic spanning conditions satisfy this property due to the unique continuity property of Čech theory. The linking number spanning conditions satisfy the property due to the fact that null intersection of compact sets is an open condition. See Definition 2.7 for more details and Fig. 2 for an illustrative example.

1.3 Methods

Our methods are those of classical geometric measure theory, drawing tools from Besicovitch [3,4,5], Reifenberg [28], Federer and Fleming [17], Federer [18], and our previous work [21, 22]. We do not use quasiminimal sets, varifolds or currents at any stage in our proof, nor do we reference any results that require their use. Our proof of rectifiability is not based on density and Preiss’s theorem, as was our isotropic result [22], but rather on Federer–Fleming and Besicovitch–Federer projections. Indeed, our axiomatic approach requires deformation versions of these projections, as well as a cone construction over a Federer–Fleming projection (Lemma 3.4) and Reifenberg’s isoperimetric inequality [26, 28].

Reifenberg regular sequences are sequences of sets $X_k\subset \mathbb {R}^n$ in which $X_k$ has a uniform lower bound on density ratios, down to a scale that decreases as k increases (see Definition 4.1). It is not too hard to produce a Reifenberg regular minimizing sequence [22]. The main result Theorem 2.5 of the current paper requires showing that two key constructions of Fleming (see Lemma 3.6 and [18] 8.2) and Almgren (see Theorem 4.13 and [1] 3.2(c)) can be applied to such Reifenberg regular minimizing sequences. Limits of Reifenberg regular sequences have nice properties, most importantly the possession of a uniform lower density bound.

2 Notation

For $p\in \mathbb {R}^n$ and $0<r<\infty$, let B(p, r) denote the closed ball of radius r about p. For $X\subset \mathbb {R}^n$,

• $X(p,r) = X \cap B(p,r)$;

• $\bar{X}$ is the closure of X;

• $\mathcal {N}(X,\epsilon )$ is the open epsilon neighborhood of X;

• $N(X,\epsilon )$ is the closed epsilon neighborhood of X;

• $\mathcal {H}^m(X)$ is the m-dimensional Hausdorff measure of X;

• $C_p(X)$ is the (inward) cone over X with basepoint p;

Moreover, let $d_H(\cdot ,\cdot )$ denote the Hausdorff distance, let $\alpha _n$ denote the Lebesgue measure of B(0, 1), and let $\mathrm {Gr}(m,n)$ denote the Grassmannian of un-oriented m-planes through the origin in $\mathbb {R}^n$, with metric $d_{\mathrm {Gr}}$.

3 Definitions and main result

Definition 2.1

A metric space C is a Lipschitz neighborhood retract if there exists an isometric embedding $C\hookrightarrow \mathbb {R}^n$ for some $n>0$, together with a neighborhood $U\subset \mathbb {R}^n$ of C and a Lipschitz retraction $\pi {:}\,U\rightarrow C$ (to simplify notation, we identify C with its image under the embedding). We say a Lipschitz neighborhood retract is localizable if there exists an embedding as above, such that for every $p\in C$ there exist $\kappa _p<\infty$ and $\xi _p>0$ such that if $0<r<\xi _p$ then there exists a Lipschitz retraction $\pi _{p,r}{:}\, C\cup B(p,r)\rightarrow C$ with $\pi _{p,r}(B(p,r))=C(p,r)$ and with Lipschitz constant $\le \kappa _p$. We call $\xi _p$ the retraction radius of C at p. If C is compact, the condition of being localizable implies that C is a priori a Lipschitz neighborhood retract. Localizability also implies local contractibility. If $\kappa :=\sup _{p \in C} \{\kappa _p\}<\infty$, then we say the localizable Lipschitz neighborhood retract is uniform.

For example, a Riemannian manifold is a uniform localizable Lipschitz neighborhood retract.³

Let C be a metric space and suppose for a moment that we have a fixed isometric embedding $C\subset \mathbb {R}^n$. For $1\le m\le n$, let $T^mC$ denote the subbundle of the restriction to C of the unoriented Grassmannian bundle $\mathbb {R}^n\times \mathrm {Gr}(m,n)\rightarrow \mathbb {R}^n$ consisting of pairs $(p,E)\in C\times \mathrm {Gr}(m,n)$ such that E is the unique approximate tangent space at p for some $\mathcal {H}^m$ measurable m-rectifiable subset X of C with $\mathcal {H}^m(X)<\infty$. Let $T_p^m C$ denote the fiber of $T^m C$ above p.

Let $0<a\le b<\infty$ and suppose $f{:}\,T^mC\rightarrow [a,b]$ is measurable (for the Borel $\sigma$-algebra on $T^mC$). For an $\mathcal {H}^m$ measurable m-rectifiable set $X\subset C$, define

$$\begin{aligned} \mathcal {F}^m(X)= \int _X f(q,T_q X) d\mathcal {H}^m, \end{aligned}$$

where $T_q X$ denotes the unique tangent m-plane to X at q.

3.1 Ellipticity

Definition 2.2

Let C be a uniform localizable Lipschitz neighborhood retract. We say $\mathcal {F}^m$ is elliptic if there exists an embedding of C into $\mathbb {R}^n$ as a uniform localizable Lipschitz neighborhood retract, such that for every $\mathcal {H}^m$ measurable m-rectifiable subset X of C with $\mathcal {H}^m(X)<\infty$, the following condition is satisfied for $\mathcal {H}^m$ almost every $p\in X$ such that X has a unique tangent m-plane E at p: If $\epsilon >0$, there exists $s>0$ such that if $0<r<s$, then

$$\begin{aligned} (f(p,E)-\epsilon )H^m(E(p,r)) \le \mathcal {F}^m(Z\cap C) + b \mathcal {H}^m (Z{\setminus } C), \end{aligned}$$

(1)

for every m-rectifiable closed set $Z\subset B(p,r)$ such that

• (a) $Z\cap \partial B(p,r) = E\cap \partial B(p,r)$; and

• (b) There is no retraction from Z onto $E\cap \partial B(p,r)$.

This definition captures Almgren’s elliptic functionals ([1] 1.2) and generalizes them to a broader class of domains. In particular, C may be a region in $\mathbb {R}^n$ with manifold boundary or a manifold with singularities (see Fig. 1). See Sect. 2.3 for more on the ellipticity condition. If C is a Riemannian manifold, and the ellipticity condition holds for a particular embedding (of Riemannian manifolds) into $\mathbb {R}^n$, then it will hold for all such embeddings.

3.2 Main result

Definition 2.3

Let C be a metric space, $m\in \mathbb {N}$ and $A\subset C$ be closed (possibly empty). If $X\subset C$, let $X^*$ denote the subset of X consisting of points $p\in X$ such that $\mathcal {H}^m(X(p,r))>0$ for all $r>0$. We say that X is reduced if $X^*=X$. We say that $X\supset A$ is a m-dimensional surface if X is closed and $X{\setminus } A$ is m-rectifiable, reduced, and $\mathcal {H}^m(X{\setminus } A)<\infty$. If $X\supset A$, let $X^\dagger$ denote the set $(X{\setminus } A)^*\cup A$.

Definition 2.4

(Axiomatic Spanning Conditions) Let $\mathcal {S}^m(C,A)$ denote a collection of m-dimensional surfaces. We say $\mathcal {S}^m(C,A)$ is a spanning collection if the following axioms hold:

• (a) If $g{:}\,C\rightarrow C$ is a Lipschitz map that fixes A and is homotopic to the identity relative to A, and if $X\in \mathcal {S}^m(C,A)$, then $g(X)^\dagger \in \mathcal {S}^m(C,A)$.

• (b) If $(X_k)_{k\in \mathbb {N}}$ is a sequence of elements of $\mathcal {S}^m(C,A)$ and $X_k \rightarrow X_0$ in the Hausdorff distance, and if $X_0{\setminus } A$ is m-rectifiable and satisfies $\mathcal {H}^m(X{\setminus } A)<\infty$, then $X_0^\dagger \in \mathcal {S}^m(C,A)$.

We will also call $\mathcal {S}^m(C,A)$ a spanning collection in the case that C is a Riemannian manifold if in place of Axiom (a), the following weaker axiom holds:

(a)$^\prime$ :

If $g{:}\, C\rightarrow C$ is a diffeomorphism that fixes A and is isotopic to the identity relative to A, and if $X\in \mathcal {S}^m(C,A)$, then $g(X)^{\dagger }\in \mathcal {S}^m(C,A).$

Our main result is the following:

Theorem 2.5

Suppose that C is a compact uniform localizable Lipschitz neighborhood retract and that $A\subset C$ is closed (possibly empty). Let $m\in \mathbb {N}$. If $\mathcal {S}^m(C,A)$ is a non-empty spanning collection and $\mathcal {F}^m$ is elliptic,⁴ then $\mathcal {S}^m(C,A)$ contains an element that minimizes the functional $X\mapsto \mathcal {F}^m(X{\setminus } A)$ among elements of $\mathcal {S}^m(C,A)$.

We now provide some examples of spanning collections.

Examples 2.6

• Algebraic spanning conditions

Definition 2.7

Suppose (C, A) is a compact pair. An algebraic spanning condition is a subset $\mathcal {L}=L_1\sqcup L_2\sqcup L_3\sqcup L_4$ of $H_{m-1}(A)\sqcup H^{m-1}(A)\sqcup H_m(C)\sqcup H^m(C)$. These are understood to be reduced Čech (co)homology groups. The coefficients may vary between the four so long as the homology groups have compact coefficients and the cohomology groups have coefficients from an $R$-module where $R$ is a commutative ring. Let $X\subset C$ be compact with $X\supset A$ and let $i{:}\,A\rightarrow X$ and $j: X\rightarrow C$ denote the inclusions. We say X spans $\mathcal {L}$ if $L_1\subset \ker (i_*)$, $L_2\cap im (i^*)=\varnothing$, $L_3\subset im (j_*)$, and $L_4\cap \ker (j^*)=\varnothing$. Let $\mathcal {S}^m(C,A,\mathcal {L})$ denote the set of all m-dimensional surfaces that span $\mathcal {L}$.

Fig. 2

The surface X described above spans the cubical frame A within the ambient space $C= B(0,1){{\setminus }}B(0,1/5)$ under a number of different algebraic spanning conditions

It follows from standard Čech theory that $\mathcal {S}^m(C,A,\mathcal {L})$ is a spanning collection. See [22] Lemmas 1.2.4 and 1.2.17, and Adams’ appendix in [28] for details.

For example, suppose $C = B(0,1) {\setminus } B(0,1/5) \subset \mathbb {R}^3$ and A is the cubical frame in Fig. 2. Let $L_1$ be any element of $H_1(A)$ and $L_3$ a generator of $H_2(C)$. Let $L_2 = L_4 = \varnothing$. Let $A'$ be the slightly curved shaded cube inside C well within the frame A. Let X be the surface depicted in Fig. 2 which consists of $A'$ union the 12 shaded trapezoids connecting the edges of $A'$ to the corresponding parallel edges of A. It follows that X is an element of $\mathcal {S}^m(C,A,\mathcal {L})$.

• Linking number spanning conditions

  If C is a Riemannian manifold, a spanning collection may be defined using the linking number spanning condition defined by the authors in [21]. The following definition generalizes this idea:

Definition 2.8

Suppose C is an n-dimensional Riemannian manifold and that $\mathcal {M}$ is a collection of compact, smoothly embedded $(n-m)$-dimensional submanifolds $M\subset C$ that is invariant under diffeomorphisms that fix A and are isotopic to the identity relative to A. Let $\mathcal {S}^m(C,A,\mathcal {M})$ be the collection of all m-dimensional surfaces X that intersect non-trivially with every element of $\mathcal {M}$. We say that elements of $\mathcal {S}^m(C,A,\mathcal {M})$ satisfy a linking number spanning condition.

It is straightforward to show that $\mathcal {S}^m(C,A,\mathcal {M})$ is a spanning collection.

• Sliding boundaries and minimizers

  One may pick some initial m-dimensional surface X, and define $\mathcal {S}^m(C,A)$ to be the smallest spanning collection that contains X, and that is also closed under the action of Lipschitz functions $g: (C,A)\rightarrow (C,A)$ that are homotopic to the identity. This is a version of what is known in the literature as sliding boundaries (see Fig. 3). Sliding boundaries and minimizers have been studied in continuum mechanics for years, (see [25], for example) and a definition for sliding boundaries suitable for geometric measure theory was introduced in [6], while [9, 11] use a somewhat different one.

Fig. 3

Sliding boundaries. In each of the three figures there are three competitors including one shaded with parallel lines and with minimal area. a Depicts the classical Plateau problem where the bounding set is a circle S. The bounding set of both figures b and c is a 2-torus

3.3 More on ellipticity

Ellipticity of $\mathcal {F}^m$ is implied in the following case: Suppose C is a Riemannian manifold. For $p\in C$, let $f_p: T^m C\rightarrow [a,b]$ denote the function $(q,T)\mapsto f(p, \tilde{T})$, where it is understood that

• (a) The function $f_p$ is only defined for q in a normal neighborhood $\mathcal {U}_p$ of p; and

• (b) The m-plane $\tilde{T}$ denotes the parallel transport of T along the unique minimal geodesic from q to p.

For an m-rectifiable $\mathcal {H}^m$ measurable set $X\subset \mathcal {U}_p$, define $\mathcal {F}_p^m(X)=\int _X f_p(q,T_q X) d\mathcal {H}^m$. For $(p,E)\in T^m C$ and $r>0$ small enough, let $S^{m-1}(p,E,r)$ denote the set of points of $\exp _p\lfloor _{\exp _p^{-1}(\mathcal {U}_p)}(E)$ which are at distance r from p. Suppose:

• (a) For all $(p,E)\in T^m C$ there exists $e(p,E)>0$ such that if $r>0$ is small enough and $D\subset \mathcal {U}_p$ is an area-minimizing m-disk with boundary $S^{m-1}(p,E,r)$, then

  $$\begin{aligned} \mathcal {F}_p^m(X)-\mathcal {F}_p^m(D)\ge e(p,E) (\mathcal {H}^m(X)-\mathcal {H}^m(D)) \end{aligned}$$

  (2)

  for all compact m-rectifiable sets $X\subset \mathcal {U}_p\cap B(p,r)$ that contain $S^{m-1}(p,E,r)$ and such that there is no Lipschitz retraction $X\rightarrow S^{m-1}(p,E,r)$ in the $C^0$ closure of the diffeomorphisms of C that fix $C{\setminus } B(p,r)$; and

• (b) The function f is equi-lower semicontinuous, in the sense that for each $p\in C$ and each $\epsilon >0$, there exists $\delta >0$ such that if $d(p,q)<\delta$ and $d_{\mathrm {Gr}}(T,\tilde{S})<\delta$, then $f(p,T)\le f(q,S)+ \epsilon$.

Then $\mathcal {F}^m$ is elliptic. Indeed, aside from a slight variation of the collection of sets X that must satisfy (2), the assumption (a) is the ellipticity condition defined in [1] 1.2.

4 Constructions

Lemma 3.1 is a slight generalization of [13] Proposition 3.1, which is a version of the Federer–Fleming projection theorem [17] 5.5, first modified for sets in [1] 2.9 (see [16] for a much simpler proof). Given a closed n-cube $Q\subset \mathbb {R}^n$ and $j \ge 0$, let $\Delta _j(Q)$ denote the collection of all n-cubes in the j-th dyadic subdivision of Q. For $0 \le d \le n$ let $\Delta _{j,d}(Q)$ denote the collection of the d-dimensional faces of the n-cubes in $\Delta _j(Q)$ and let $S_{j,d}(Q)\subset Q$ denote the set union of these faces.

Lemma 3.1

Suppose E is a closed subset of Q such that $\mathcal {H}^d(E) < \infty$. For each $j \ge 0$, there exists a Lipschitz map $\phi : \mathbb {R}^n\times [0,1] \rightarrow \mathbb {R}^n$ with the following properties:

• (a) $\phi _t = Id$ on $(\mathbb {R}^n{\setminus } Q)\cup S_{j,d}(Q)$ for all $t\in [0,1]$;

• (b) $\phi _0 = Id$;

• (c) $\phi _1(E) \subset S_{j,d}(Q) \cup \partial Q$;

• (d) $\phi (R\times [0,1]) \subset R$ for each $R \in \Delta _j(Q)$;

• (e) $\mathcal {H}^d(\phi _1(E \cap R)) \le c_1 \mathcal {H}^d(E \cap R)$ for all $R \in \Delta _j(Q)$ where $c_1=c_1(n)$ depends only on n;

• (f) $\mathcal {H}^{d+1}(\phi (E\cap R\times [0,1]))\le c_1 2^{-j}diam (Q) \mathcal {H}^d(E\cap R)$ for all $R\in \Delta _j(Q)$;

• (g) If E is K-semiregular,⁵ then the Lipschitz constant of $\phi _1$ depends only on n and K. Furthermore, there exists a constant $\delta >0$ depending only on n and K such that if $F\subset Q\cap \mathcal {N}(E,\delta 2^{-j}diam (Q))$, then $\phi _1(F)\subset S_{j,d}(Q)\cup \partial Q$.

Proof

The map $\phi$ is the concatenation of the straight line homotopies between the maps $\psi _m$ in [13] Lemma 3.10. These maps $\psi _m$ consist, roughly speaking, of radial projections onto $S_{j,m}(Q)\cup \partial Q$ away from points near the center of each $(m+1)$-face $F\in \Delta _{j,m+1}(Q)$.

Parts (a)–(e) are [13] Proposition 3.1. Part (f) is then apparent from [28] Lemma 6 and [13] (3.14) Part (g) follows from [13] Lemma 3.31 and (3.33) $\square$

A trivial yet important fact is the following:

Lemma 3.2

If $(X_i)_{i\in \mathbb {N}}$ is a sequence of compact subsets of $\mathbb {R}^n$ and $X_i$ converges to a compact set X in the Hausdorff metric, and if $f: \mathbb {R}^n\rightarrow \mathbb {R}^n$ is continuous, then $f(X_i)$ converges to f(X) in the Hausdorff metric.

Suppose $X\subset \mathbb {R}^n$ is closed and $p\in \mathbb {R}^n$. We shall repeatedly make use of a “cone construction” where X is replaced by a subset $\tilde{X}$ of $(X {\setminus } B(p,r)) \cup C_p(X\cap \partial B(p,r))$. Specifically, $\tilde{X}$ will be a Hausdorff limit of the images of X under a sequence of diffeomorphisms that leave $\mathbb {R}^n {\setminus } B(p,r)$ fixed. These diffeomorphisms will be defined so that they squeeze an increasing portion of B(p, r) down successively closer to p. This construction is due to [9]. However, the Hausdorff measure of the resulting set $\tilde{X} \cap B(p,r)$ does not have good estimates unless $X\cap \partial B(p,r)$ is polyhedral (see [28] Lemmas 5 and 6.) Instead, we will use a Federer–Fleming projection to push $X\cap \partial B(p,r)$ onto a cubical grid before coning, and this will yield nicer estimates (see Lemma 3.4.)

Lemma 3.3

Suppose $X\subset \mathbb {R}^n$ is compact, $p\in \mathbb {R}^n$ and $r>0$. There exists a sequence of diffeomorphisms $\xi _i$ of $\mathbb {R}^n$ that are the identity outside B(p, r), such that $\xi _i(X)$ converges in the Hausdorff metric to a compact set Y that is contained in $X{\setminus } B(p,r) \cup C_p(X\cap \partial B(p,r))$.

Proof

For $t\in [0,1)$, let $\psi _t:[0,\infty )\rightarrow [0,\infty )$ be a smooth, increasing function, such that

• (a) $\psi _t'(s)>0$ for all s;

• (b) $\psi _t(rt)< r(1-t)$;

• (c) $\psi _t(s)=s$ for all $s\ge r$.

Pick some sequence $(t_i)_{i\in \mathbb {N}}\subset [0,1)$ with $t_i\rightarrow 1$ and let $\xi _i(x) = p + \psi _{t_i}(|x-p|)\frac{x-p}{|x-p|}$ for $x \in \mathbb {R}^n$. Then $\xi _i$ is a diffeomorphism of $\mathbb {R}^n$ satisfying $\xi _i = Id$ on $\mathbb {R}^n{\setminus } B(p,r)$ for all $i\in \mathbb {N}$, and $\lim _{i\rightarrow \infty } \xi _i(x) = p$ for each $x\in \mathcal {N}(p,r)$.

By taking a subsequence if necessary, we may assume without loss of generality that $(\xi _i(X))_{i\in \mathbb {N}}$ converges in the Hausdorff metric to a compact set Y. By construction, this set Y is contained in $X{\setminus } B(p,r) \cup C_p(X\cap \partial B(p,r))$. $\square$

Lemma 3.4

Suppose (C, X, A) is a compact triple, C is a uniform localizable Lipschitz retract, and $X\in \mathcal {S}^m(C,A)$. Let $p\in C{\setminus } A$ and suppose $r>0$ is chosen smaller than the retraction radius of C at p, and so that $B(p,\sqrt{n}r)\cap A=\varnothing$, $X\cap \partial B(p,r)$ is $(m-1)$-rectifiable, and $\mathcal {H}^{m-1}(X\cap \partial B(p,r))<\infty$. Then for every $\epsilon >0$ there exist compact sets $P_\epsilon \subset C(p,r)$ and $T_\epsilon \subset C(p,r)$ such that

• (a) $T_\epsilon \subset \mathcal {N}(X\cap \partial B(p,r), \epsilon r)$;

• (b) $\mathcal {H}^m(T_\epsilon ) \le c_1 \epsilon r \mathcal {H}^{m-1}(X\cap \partial B(p,r))$;

• (c) $\mathcal {H}^m(P_\epsilon ) \le \gamma r^m$ where $0<\gamma <\infty$ depends on n, C and $\epsilon$;

• (d) $(P_\epsilon \cup T_\epsilon \cup (X {\setminus } B(p,r)))^\dagger$ is an element of $\mathcal {S}^m(C,A)$.

Proof

By Lemma 3.3, the set $X{\setminus } B(p,r) \cup C_p(X\cap \partial B(p,r))$ contains a Hausdorff limit Y of deformations of X by diffeomorphisms of $\mathbb {R}^n$ that fix $\mathbb {R}^n{\setminus } B(p,r)$. We shall deform Y using a modification of 3.1 and construct for each $0<\delta <1$ a Lipschitz deformation of B(p, r) that maps each sphere $\partial B(p,s)$ to itself for $0 < s \le r$, maps radial rays to radial rays on $B(p,(1-\delta )r)$ and is the identity on $\partial B(p,r)$.

Let $\Pi _r: \mathbb {R}^n{\setminus } \{p\} \rightarrow \partial B(p,r)$ and $\Pi _{r,s}: \partial B(p,r) \rightarrow \partial B(p,s)$ denote radial projections.

Let Q be an n-cube of side length 2r centered at p and apply Lemma 3.1 to $d = m-1$ and $E = Y\cap \partial B(p,r)$ and for a fixed number j of subdivisions of Q, to be determined in a moment. Obtain the Federer–Fleming map $\phi :\mathbb {R}^n \times [0,1] \rightarrow \mathbb {R}^n$ from Lemma 3.1 and let $\tilde{\phi }_t = \Pi _r \circ \phi _t$. Since $Y\cap \partial B(p,r)$ is $(m-1)$-rectifiable, so is $\tilde{\phi }_t(Y\cap \partial B(p,r))$. So, for each $t\in [0,1]$, the map $\tilde{\phi }_t$ restricts to a map from B(p, r) to itself. At $t=0$ the map is the identity and at $t=1$, it sends $Y\cap \partial B(p,r)$ to $\Pi _r(\phi _1(Y\cap \partial B(p,r)))$.

Using this homotopy, we shall define a Lipschitz map $\psi : \mathbb {R}^n \rightarrow \mathbb {R}^n$ so that $\psi$ sends each sphere $\partial B(p,s)$ to itself for $0 < s \le r$. Suppose $(1 - \delta )r \le s \le r$.

Let

$$\begin{aligned} \psi \lfloor _{\partial B(p,s)} = \Pi _{r,s} \circ \tilde{\phi }_{(r-s)/(\delta r)} \circ \Pi _{r,s}^{-1}. \end{aligned}$$

Extend $\psi$ to $B(p, r - \delta r)$ in the unique way such that each ray from p to $q \in \partial B(p, r - \delta r)$ is mapped to the ray from p to $\psi (q)$ and so that $\psi (B(p,s))\subset B(p,s)$ for each $0\le s<(1-\delta )r$. Finally, extend $\psi$ to the identity on $\mathbb {R}^n{\setminus } B(p,r)$.

Let $\pi _{p,r}:\mathbb {R}^n\rightarrow C(p,r)$ denote the Lipschitz retraction given by Definition 2.1. It follows from Lemma 3.1 (e) that we may choose j large enough and $0<\delta <1$ small enough so that

$$\begin{aligned} \tilde{T_\epsilon } := \pi _{p,r} (\psi (Y(p,r){\setminus } \mathcal {N}(p,(1-\delta )r))) \end{aligned}$$

satisfies (b) and (a). The set $T_\epsilon$ will be defined as a subset of $\tilde{T_\epsilon }$.

Likewise, let

$$\begin{aligned} \tilde{P_\epsilon } = \pi _{p,r} (\psi (Y(p, (1-\delta )r))). \end{aligned}$$

We will define $P_\epsilon$ as a subset of $\tilde{P_\epsilon }$, so let us establish (c): Let $N_\epsilon$ be an upper bound on the number of $(m-1)$-dimensional faces of a cubical grid of side length $2^{-j(\epsilon )}$ within $\epsilon$ of $\partial B(0,1)$. Since

$$\begin{aligned} \phi _1(X\cap \partial B(p,r)) \subset \mathcal {N}(X\cap \partial B(p,r),\epsilon r)\cap S_{j,m-1}(Q), \end{aligned}$$

it follows from [28] Lemmas 2 and 5 that

$$\begin{aligned} \mathcal {H}^m(P_\epsilon )&\le \frac{\kappa ^m r\sqrt{n} N_\epsilon }{m} \left( \frac{r}{2^{j+1}}\right) ^{m-1}\\&\le \kappa ^m \sqrt{n} N_\epsilon \left( \frac{\epsilon }{2\sqrt{n}}\right) ^{m-1} r^m. \end{aligned}$$

We know that for each i, the map $\pi _{p,r}\circ \psi \circ \xi _i:C\rightarrow C$ is Lipschitz and homotopic to the identity relative to A (indeed, the map fixes the complement of a ball that misses A) and so by Axiom (a), we know that $(\pi _{p,r}\circ \psi \circ \xi _i (X))^\dagger \in \mathcal {S}^m(C,A)$. By Lemma 3.2, we know that $\pi _{p,r}\circ \psi \circ \xi _i (X)$ converges in the Hausdorff metric to $\tilde{Z}=\tilde{P_\epsilon } \cup \tilde{T_\epsilon } \cup (X {\setminus } B(p,r))$, which by construction is m-rectifiable away from A and satisfies $\mathcal {H}^m(Z{\setminus } A)<\infty$. By taking a subsequence if necessary, $(\pi _{p,r}\circ \psi \circ \xi _i (X))^\dagger$ converges in the Hausdorff metric to a subset Z of $\tilde{Z}$ that contains $X{\setminus } B(p,r)$. Let $P_\epsilon =\tilde{P_\epsilon }\cap Z$ and let $T_\epsilon =\tilde{T_\epsilon }\cap Z$. Thus (d) holds by Axiom (b).

Now if C is a Riemannian manifold and Axiom (a)$^\prime$ holds in place of Axiom (a), then we proceed as above in a coordinate chart, but omit the map $\pi _{p,r}$. $\square$

Lemma 3.5

Suppose we are given a closed subset E of an n-cube $Q\subset \mathbb {R}^n$ such that $\mathcal {H}^d(E) < \infty$, and an $\mathcal {H}^d$ measurable and purely d-unrectifiable set $U\subset E$ whose closure is contained in the interior of Q. Then for $0\le j<\infty$ large enough, there exists a map $\phi$ satisfying Lemma 3.1 (b)–(f), such that

$$\begin{aligned} \mathcal {H}^d(\phi _1(U))=0 \end{aligned}$$

(3)

and

$$\begin{aligned} \phi _t = Id \text { on } \mathbb {R}^n{\setminus } Q \quad \text { for all } t\in [0,1]. \end{aligned}$$

(4)

Proof

We construct the map $\phi$ as follows. Choose j large enough so that U is contained entirely within the subset of cubes in $\Delta _j(Q)$ that do not have any faces on the boundary of Q. Within those cubes, approximate the maps $\psi _k$, $d \le k < n$ defined in the proof of [13] Proposition 3.1 by diffeomorphisms $\tilde{\psi }_k$. For each $d\le k< n$, the image of U by the map $\tau _k \equiv \tilde{\psi }_k\circ \cdots \circ \tilde{\psi }_{n-1}$ remains purely d-unrectifiable and $\tau _k(E)$ will be contained in an open $\epsilon$-neighborhood of $\Delta _{j,k}(Q)$.

The map $\phi _1$ is the composition $\phi _1\equiv \rho \circ \theta \circ \tau _d$, where the maps $\rho$ and $\theta$ are defined below. The map $\phi$ is defined as in Lemma 3.1 to be the concatenation of the straight-line homotopies between the maps $\psi _n, \tilde{\psi }_{n-1}, \dots , \tilde{\psi }_d, \theta$, and $\rho$.

By the Besicovitch–Federer projection theorem, for each $0<\delta <\epsilon$ and each d-face $F\in \Delta _{j,d}(Q)$ there exists an affine d-plane $\tilde{F}$ such that $F\subset \mathcal {N}(\tilde{F},\delta )$, and such that the image of $\phi _1(U)$ by the orthogonal projection $\Theta _{\tilde{F}}: \mathbb {R}^n \rightarrow \tilde{F}$ has zero Hausdorff d-measure.

Let $\zeta >2\epsilon$ and define $\theta$ on $\mathcal {N}(F, \epsilon ){\setminus } \mathcal {N}(\partial F, \zeta )$ to be the map $\Theta _F\circ \Theta _{\tilde{F}}$, where $\Theta _F$ denotes orthogonal projection onto the affine d-plane defined by F. We may extend $\theta$ as a Lipschitz map on $\mathcal {N}(\Delta _{j,d}(Q), \epsilon )\cap N(\Delta _{j,d-1}(Q),\zeta )$ so that

$$\begin{aligned} \theta (\mathcal {N}(\Delta _{j,d}(Q), \epsilon )\cap N(\Delta _{j,d-1}(Q),\zeta )) \subset \Delta _{j,d}(Q)\cap \mathcal {N}(\Delta _{j,d-1}(Q),2\zeta ), \end{aligned}$$

(5)

and so that the Lipschitz constant of $\theta$ depends only on n. Finally, extend $\theta$ to $\mathbb {R}^n$ as a Lipschitz map.

The map $\rho$ is defined on each face $F\in \Delta _{j,d}(Q)$ as follows: Let q be the center point of F and let $\chi _{q,\zeta }(x)$ denote the point

$$\begin{aligned} \frac{1}{1-2\sqrt{d}\zeta }(x-q)+q. \end{aligned}$$

For $x\in F{\setminus } \{q\}$, let $\omega (x)$ denote the point on $\partial F$ and the ray passing through x and ending at q. For $x\in F$, let

$$\begin{aligned} \rho (x)=\left\{ \begin{array}{ll} \chi _{q,\zeta }(x), &{}\quad \text {if } \chi _{q,\zeta }(x)\in F\\ \omega (x), &{}\quad \text {otherwise.} \end{array}\right. \end{aligned}$$

Then $\rho$ is Lipschitz with Lipschitz constant close to 1 (controlled by $\zeta$.) Finally, extend $\rho$ to $\mathbb {R}^n$, with proportional Lipschitz constant.

Note that by (5), $\mathcal {H}^d$ almost all $\theta (U)$ is contained in $\Delta _{j,d}(Q)\cap \mathcal {N}(\Delta _{j,d-1}(Q), 2\zeta )$, and this region is collapsed onto $\Delta _{j,d-1}(Q)$ by $\rho$. Thus, (3) holds. It is apparent that (4) holds, as well as Lemma 3.1 (b)–(c). To see (e) and (f), it is enough to observe that [13] (3.20) still holds for the modified maps $\tilde{\psi }_k$. Finally, (g) holds since [13] (3.33) still holds for the modified maps. $\square$

Lemma 3.6

(Upper bounds on density ratios) Suppose $Y_k\subset \mathbb {R}^n$ for $k \ge 1$. Fix $p \in \mathbb {R}^n$, $0< r< R <\infty$ and $0\le \eta \le \infty$. Suppose

$$\begin{aligned} \limsup _{k\rightarrow \infty } \frac{ \mathcal {H}^m(Y_k(p,R))}{R^m} \le \eta \end{aligned}$$

and that for each $\delta >0$ there exists $M_\delta$ such that if $k \ge M_\delta$, then

$$\begin{aligned} \mathcal {H}^m(Y_k(p,s)) < \frac{s}{m} \frac{d}{ds}\mathcal {H}^m(Y_k(p,s)) + \delta \end{aligned}$$

(6)

for almost every $s\in [r, R)$ satisfying $\mathcal {H}^m(Y_k(p,s))/s^m \ge \eta$. Then

$$\begin{aligned} \limsup _{k\rightarrow \infty } \frac{ \mathcal {H}^m(Y_k(p,r))}{r^m} \le \eta . \end{aligned}$$

Proof

If $\eta =\infty$ or $\eta =0$ there is nothing to prove, so let us assume $0<\eta <\infty$. Suppose

$$\begin{aligned} \limsup _{k\rightarrow \infty } \frac{ \mathcal {H}^m(Y_k(p,r))}{r^m} > \eta . \end{aligned}$$

Let $\epsilon > 0$ and

$$\begin{aligned} 0< \delta < \min \left\{ \epsilon , \left( \limsup _{k \rightarrow \infty } \frac{\mathcal {H}^m(Y_k(p,r))}{r^m} - \eta \right) \right\} \frac{r^{m+1}}{mR}. \end{aligned}$$

(7)

Let $k_i \rightarrow \infty$ with $k_i>M_\delta$,

$$\begin{aligned}&\left( \frac{\mathcal {H}^m(Y_{k_i}(p,r))}{r^m} - \eta \right) \frac{r^{m+1}}{mR}> \delta , \end{aligned}$$

(8)

$$\begin{aligned}&\mathcal {H}^m(Y_{k_i}(p,r)) \rightarrow \limsup _{k \rightarrow \infty } \mathcal {H}^m(Y_k(p,r)), \end{aligned}$$

(9)

and

$$\begin{aligned} \frac{\mathcal {H}^m(Y_{k_i}(p,r))}{r^m} > \eta . \end{aligned}$$

Let $J_i = [r,r_i)$ be the largest half-open interval with right endpoint $r_i\in (r,R]$ such that $\mathcal {H}^m(Y_{k_i}(p,t))/t^m \ge \eta$ for almost every $t \in J_i$. By (8),

$$\begin{aligned} \frac{\mathcal {H}^m(Y_{k_i}(p,r))}{r^m} > \eta + \frac{m}{r^{m+1}} \delta (r_i-r). \end{aligned}$$

(10)

By (6), we have

$$\begin{aligned} \frac{d}{dt} \frac{\mathcal {H}^m(Y_{k_i}(p,t))}{t^m} = \frac{t \frac{d}{dt}(\mathcal {H}^m(Y_{k_i}(p,t))) - m \mathcal {H}^m(Y_{k_i}(p,t)))}{t^{m+1}} > -\frac{m \delta }{t^{m+1}} \ge - \frac{m\delta }{r^{m+1}} \end{aligned}$$

for almost every $t\in J_i$.

Integrating yields,

$$\begin{aligned} -\frac{m\delta (r_i-r)}{r^{m+1}} < \int _r^{r_i} \frac{d}{dt}\left( \frac{\mathcal {H}^m(Y_{k_i}(p,t))}{t^m}\right) dt \le \frac{\mathcal {H}^m(Y_{k_i}(p,r_i))}{r_i^m} - \frac{\mathcal {H}^m(Y_{k_i}(p,r))}{r^m} \end{aligned}$$

(11)

Combining (10) and (11) yields

$$\begin{aligned} \frac{\mathcal {H}^m(Y_{k_i}(p,r_i))}{r_i^m}> \eta . \end{aligned}$$

It follows that $r_i = R$. Since $\delta \le \epsilon r^{m+1}/(m R)$, (11) implies

$$\begin{aligned} \frac{\mathcal {H}^m(Y_{k_i}(p,r))}{r^m} < \frac{\mathcal {H}^m(Y_{k_i}(p,R))}{R^m} + \epsilon . \end{aligned}$$

(12)

Letting $i\rightarrow \infty$ and then $\epsilon \rightarrow 0$ we deduce from (9) that

$$\begin{aligned} \limsup _{k \rightarrow \infty }\frac{\mathcal {H}^m(Y_{k_i}(p,r))}{r^m} \le \eta , \end{aligned}$$

a contradiction. $\square$

For $E\in \mathrm {Gr}(m,n)$, $p\in \mathbb {R}^n$ and $0<\epsilon <1$, let $\mathcal {C}(p,E,\epsilon ) = \{q \in \mathbb {R}^n: d_H(\{q-p\}, E) < \epsilon \Vert p-q\Vert \}$.

Lemma 3.7

Suppose we are given $X\subset \mathbb {R}^n$, a point $p\in X$, and $s > 0$ such that

$$\begin{aligned} \inf \left\{ \frac{\mathcal {H}^m(X(q,r))}{r^m}: q\in X, B(q,r)\subset B(p,s) \right\} >0. \end{aligned}$$

If $E\in \mathrm {Gr}(m,n)$ is an approximate tangent m-plane for X at p, then for every $0<\epsilon <1$ there exists $r>0$ such that $X(p,r){\setminus } \overline{\mathcal {C}(p,E,\epsilon )}=\varnothing$.

Proof

If the result is false, then for some $0<\epsilon <1$ there exist sequences $r_i \rightarrow 0$ and $q_i \in X(p,r_i) {\setminus } \overline{\mathcal {C}(p,E,\epsilon )}$. Let $s_i = 2\Vert p-q_i\Vert$. Then

$$\begin{aligned} B(q_i,\epsilon s_i/4) \subset B(p,s_i) {\setminus } \mathcal {C}(p,E,\epsilon /4) \end{aligned}$$

and thus

$$\begin{aligned} \frac{\mathcal {H}^m(X(p,s_i){\setminus } \mathcal {C}(p,E,\epsilon /4))}{s_i^m } \ge \frac{\mathcal {H}^m(X(q_i,\epsilon s_i/4))}{s_i^m}, \end{aligned}$$

the right hand side of which for large enough i is bounded below by

$$\begin{aligned} (\epsilon /4)^m \inf \left\{ \frac{\mathcal {H}^m(X(q,r))}{r^m}: q\in X,\quad \text { and } B(q,r)\subset B(p,s) \right\} > 0, \end{aligned}$$

a contradiction. $\square$

5 Minimizing sequences

We now begin the proof of Theorem 2.5. Let us assume for the remainder of the paper that the conditions of Theorem 2.5 are satisfied, namely that C is a compact uniform localizable Lipschitz neighborhood retract, that $A\subset C$ is closed, that $m\in \mathbb {N}$, that $\mathcal {F}^m$ is elliptic, and that $\mathcal {S}^m(C,A)$ is a non-empty spanning collection. Fix an embedding $C\hookrightarrow \mathbb {R}^n$ as a uniform localizable Lipschitz neighborhood retract and let $\pi : U\rightarrow C$ be a Lipschitz retraction onto C of an open neighborhood U of C.

By Lemma 3.1 there exists a sequence $(X_k)_{k\in \mathbb {N}}$ of elements of $\mathcal {S}^m(C,A)$ such that

• (a) $\mathcal {F}^m(X_k{\setminus } A) \rightarrow \mathfrak {m} := \inf \{\mathcal {F}^m(Y{\setminus } A):Y \in \mathcal {S}^m(C,A) \}$;

• (b) The sequence of Borel measures $(\mathcal {F}^m\lfloor _{X_k{\setminus } A})_{k\in \mathbb {N}}$ on $\mathbb {R}^n$ converges weakly to a finite Borel measure $\mu _0$ on $\mathbb {R}^n$; and

• (c) The sequence $(X_k)_{k\in \mathbb {N}}$ converges to $X_0 \equiv {supp}(\mu _0)\cup A$ in the Hausdorff metric.

Indeed, consider a sequence $X_k$ satisfying properties (a) and (b). Such a sequence exists by weak compactness. Consider an n-cube $Q\subset \mathbb {R}^n$ that contains C. Let $(e_i)_{i\in \mathbb {N}}$ be a sequence of positive real numbers, with $e_i\rightarrow 0$. For each $i\in \mathbb {N}$ choose $j<\infty$ large enough so that the n-cubes in the j-th dyadic subdivision $\Delta _j(Q)$ of Q that intersect $C{\setminus } \mathcal {N}(X_0,\epsilon _i)$ non-trivially, or are adjacent to such an n-cube, are contained in U and are disjoint from $\mathcal {N}(X_0,\epsilon _i/2)$. Label the collection of these n-cubes $\Gamma _i$. We will call an n-cube G of $\Gamma _i$ an interior n-cube if G is contained in the interior of the union of the elements of $\Gamma _i$.

For each $k\in \mathbb {N}$ and $i\in \mathbb {N}$, let $\phi _k^i$ be a Federer–Fleming map of $X_k$ onto the m-dimensional faces of the interior n-cubes of $\Gamma _i$. In particular, the maps $\phi _k^i$ differ from the identity only on union of the cubes of $\Gamma _i$. By property (b), it holds that for k large enough (depending on i,)

$$\begin{aligned} \mathcal {H}^m(\phi _k^i(X_k{\setminus } \mathcal {N}(X_0,\epsilon /2))) < \mathcal {H}^m(F) \end{aligned}$$

for any m-face $F\in \Delta _{j,m}(Q)$ of a cube in $\Delta _j(Q)$. Thus, for each such m-face F of an interior cube of $\Gamma$, we may radially project $\phi _k^i(X_k)$ onto $\partial F$ away from a point in the non-empty set $F{\setminus } \phi _k^i(X_k)$, an $(m-1)$-face. Let $\tilde{X}_k^i$ denote the projection $\pi$ back onto C of the resulting set. In the case that C is a Riemannian manifold and $\mathcal {S}^m(C,A)$ satisfies Axiom (a)$^\prime$ in place of Axiom (a), apply the above construction in charts and omit the map $\pi$.

Thus, in either case, the set $(\tilde{X}_k^i)^\dagger$ is an element of $\mathcal {S}^m(C,A)$ for all $i\in \mathbb {N}$ and for all $k\in \mathbb {N}$ large enough (depending on i.) It is not hard to show that there exists a diagonal subsequence⁶ $(\tilde{X}_k)_{k\in \mathbb {N}}$ of $((\tilde{X}_k^i)^\dagger )$ such that

$$\begin{aligned} \mathcal {F}^m(\tilde{X}_k{\setminus } A) \rightarrow \mathfrak {m}, \end{aligned}$$

the sequence of measures $(\mathcal {F}^m\lfloor _{\tilde{X}_k {\setminus } A})_{k\in \mathbb {N}}$ still converges weakly to $\mu _0$, and in addition the sequence $(\tilde{X}_k)_{k\in \mathbb {N}}$ converges to $X_0$ in the Hausdorff metric.

We will need a minimizing sequence that satisfies additional properties. Namely, we want one that is “Reifenberg regular,” defined as follows:

Definition 4.1

If C is a metric space and $W\subset C$, and if $0<\mathbf {c}<\infty$ and $0<\mathbf {R}\le \infty$, we say that a sequence $(X_k)_{k\in \mathbb {N}}$ of subsets of C is Reifenberg $(\mathbf {c},\mathbf {R})$-regular in W (in dimension m) if whenever we are given $k\ge 1$, a point $p\in X_k$ and a radius $2^{-k}<r< \mathbf {R}$ such that the ball B of radius r about p is contained in W, then

$$\begin{aligned} \frac{\mathcal {H}^m(X_k \cap B)}{r^m}\ge \mathbf {c}. \end{aligned}$$

If $m>1$, it follows from [22] Lemma 4.2.3 and [26] Theorem 3 that there exists a subsequence of $(X_k)_{k\in \mathbb {N}}$ that is Reifenberg $(\mathbf {c},\mathbf {R})$-regular in $C{\setminus } A$ for some $0<\mathbf {c}<\infty$ and $0<\mathbf {R}< \infty$. We now show that this is still the case if $m=1$:

Lemma 4.2

If $m=1$, then the sequence $(X_k)_{k\in \mathbb {N}}$ of elements of $\mathcal {S}^1(C,A)$ has a subsequence that is Reifenberg (1 / 4, 1)-regular in $C{\setminus } A$.

Proof

If not, then by the Eilenberg inequality [14], there exist $N_1<\infty$ and $N_2>0$ such that for all $k\ge N_1$, there exist $p_k\in X_k$ and

$$\begin{aligned} r_k\in (2^{-(N_2+1)},1) \end{aligned}$$

(13)

such that $B(p_k,r_k)$ is disjoint from A, and

$$\begin{aligned} X_k \cap \partial B(p_k,r_k) = \varnothing . \end{aligned}$$

(14)

Since $X_k$ converges to $X_0$ in the Hausdorff metric, there exists a point $p\in X_0$ and a subsequence $p_{k_i}\rightarrow p$. By (13), the distance from $p_k$ to A is bounded below, hence $p\notin A$. Assume without loss of generality that in addition, $r_{k_i}\rightarrow r$ for some $r\in [2^{-(N_2+1)},1].$ Since $p\in {supp}(\mu _0)$, we know that $\mu _0(p,r/2)>0$, and thus $\mathcal {F}^1(X_{k_i}(p_{k_i}r_{k_i}))$ has a positive lower bound for i large enough.

On the other hand, by (14), there exists for each $i\in \mathbb {N}$ a sequence of diffeomorphisms of $\mathbb {R}^n$ (or of C if C is a manifold) that fix $B(p_{k_i},r_{k_i})^c$ such that the Hausdorff limit $\tilde{X}_{k_i}$ of $X_{k_i}$ under these diffeomorphisms exists and such that $\tilde{X}_{k_i}(p_{k_i},r_{k_i})=\{p_{k_i}\}$. Thus,

$$\begin{aligned} \tilde{X}_{k_i}^\dagger \in \mathcal {S}^1(C,A) \end{aligned}$$

and

$$\begin{aligned} \mathcal {F}^1(\tilde{X}_{k_i}^\dagger ) < \mathfrak {m} \end{aligned}$$

for i large enough, a contradiction. $\square$

So, for the rest of the paper, let us fix a sequence $(X_k)_{k\in \mathbb {N}}$ of surfaces $X_k\in \mathcal {S}^m(C,A)$ such that

• (a) The sequence $( \mathcal {F}^m(X_k{\setminus } A))_{k\in \mathbb {N}}$ converges to $\mathfrak {m}$;

• (b) The sequence of Borel measures $(\mathcal {F}^m\lfloor _{X_k{\setminus } A})_{k\in \mathbb {N}}$ on $\mathbb {R}^n$ converges weakly to a finite Borel measure $\mu _0$ on $\mathbb {R}^n$;

• (c) The sequence $(X_k)_{k\in \mathbb {N}}$ converges to $X_0 \equiv {supp}(\mu _0)\cup A$ in the Hausdorff metric; and

• (d) The sequence $(X_k)_{k\in \mathbb {N}}$ is Reifenberg regular in $C{\setminus } A$.

See [22] §4.3 for the general properties of limits of Reifenberg regular sequences. In particular, by [22] Corollary 4.3.5,

$$\begin{aligned} \frac{\mu _0(B(p,r))}{r^m}\ge a\mathbf {c}>0 \end{aligned}$$

(15)

for all $p\in X_0{\setminus } A$, radii $0<r<\min \{ d_H(\{p\},A),\mathbf {R} \}$ and $m \ge 1$. Thus, there is a uniform lower bound on lower density of $\mu _0$: ${\Theta _*}^m(\mu _0,p)\ge a\mathbf {c}\alpha _m^{-1}>0$. We now show there is an upper bound for $\frac{\mu _0(B(p,r))}{r^m}$, uniform away from A.

Definition 4.3

For $p \in C{\setminus } A$ let $d_p=\min \{ d_H(\{p\},A\cup (\mathbb {R}^n{\setminus } U)),\mathbf {R}, \xi _p \}$. Let $D_p$ be the subset of $(0,d_p)$ consisting of numbers r such that the following conditions hold for all $s \in \{ qr: q\in (0,1]\cap \mathbb {Q} \}$ and $k\ge 1$:

• (a) $\mu _0(\partial B(p,s)) = 0$,

• (b) $X_k\cap \partial B(p,s)$ is $(m-1)$-rectifiable,

• (c) $\mathcal {H}^{m-1}(X_k\cap \partial B(p,s))<\infty$,

• (d) $s\mapsto \mathcal {H}^m(X_k(p,s))$ is differentiable at s, and

• (e)
  $$\begin{aligned} \lim _{h\rightarrow 0}\frac{1}{h}\int _s^{s+h}\mathcal {H}^{m-1}(X_k\cap \partial B(p,t))dt = \mathcal {H}^{m-1}(X_k\cap \partial B(p,s)). \end{aligned}$$

Lemma 4.4

The set $D_p$ is a full Lebesgue measure subset of $(0,d_p)$.

Proof

Part (b) determines a full Lebesgue measure set since $X_k$ is m-rectifiable. Part (c) follows from the Eilenberg inequality [14]. Part (d) follows since $\mathcal {H}^m(X_k(p,s))$ is monotone non-decreasing. Part (e) follows from the Lebesgue Differentiation Theorem. $\square$

Lemma 4.5

If $r \in D_p$, then $\mathcal {H}^{m-1}(X_k\cap \partial B(p,r)) \le \frac{d}{dr}\mathcal {H}^m(X_k(p,r))$ for all $k\ge 1$.

Proof

By the Eilenberg inequality [14] and Definition 4.3,

$$\begin{aligned} \mathcal {H}^{m-1}(X_k\cap \partial B(p,r))&=\lim _{h\rightarrow 0} \frac{\int _r^{r+h} \mathcal {H}^{m-1}(X_k\cap \partial B(p,t))\,dt}{h}\\&\le \lim _{h\rightarrow 0} \frac{\mathcal {H}^m(X_k(p,r+h){\setminus } X_k(p,r))}{h}\\&= \frac{d}{dr}\mathcal {H}^m(X_k(p,r)) \end{aligned}$$

$\square$

Let $M<\infty$ be an upper bound for $\{\mathcal {F}^m(X_k{\setminus } A) : k\ge 1\}$.

Lemma 4.6

Let $p \in X_0 {\setminus } A$ and $0<r\le d_p$. For each $\delta > 0$ there exists $N_{p,r,\delta } > 1$ such that if $k\ge N_{p,r,\delta }$, and $Y_k\in \mathcal {S}^m(C,A)$ satisfies

$$\begin{aligned} Y_k=(X_k{\setminus } \mathcal {N}(p,r)) \cup Z_k \end{aligned}$$

(16)

for some $\mathcal {H}^m$ measurable m-rectifiable set $Z_k\subset C$, then

$$\begin{aligned} \mathcal {F}^m(X_k \cap \mathcal {N}(p,r)) < (1+\delta )\mathcal {F}^m(Z_k {\setminus } A). \end{aligned}$$

(17)

If, in addition $0<s<r$, $\delta \le 1$ and $Z_k{\setminus } B(p,s)=(X_k\cap \mathcal {N}(p,r)){\setminus } B(p,s)$, then

$$\begin{aligned} \mathcal {F}^m(X_k(p,s)) \le 2 \mathcal {F}^m(Y_k(p,s)) + \delta M. \end{aligned}$$

(18)

Proof

If (17) fails, there exist $k_i \rightarrow \infty$ and $Y_{k_i}\in \mathcal {S}^m(C,A)$ satisfying (16) such that

$$\begin{aligned} \mathcal {F}^m(Z_{k_i}{\setminus } A) \le \frac{\mathcal {F}^m(X_{k_i}\cap \mathcal {N}(p,r)}{(1+\delta )}. \end{aligned}$$

It follows from [22] Proposition 4.3.2 that

$$\begin{aligned} \liminf _{i\rightarrow \infty }\{\mathcal {F}^m(Y_{k_i}{\setminus } A) \}&\le \liminf _{i\rightarrow \infty }\{\mathcal {F}^m(Z_{k_i}{\setminus } A) + \mathcal {F}^m(X_{k_i}{\setminus } (\mathcal {N}(p,r)\cup A)) \} \\&\le \liminf _{i\rightarrow \infty }\{\mathcal {F}^m(X_{k_i}{\setminus } A) - \frac{\delta }{1+\delta }\,\mathcal {F}^m(X_{k_i}\cap \mathcal {N}(p,r)) \} \\&\le \liminf _{i\rightarrow \infty }\{\mathcal {F}^m(X_{k_i}{\setminus } A) - \frac{a \delta }{1+\delta }\, \mathcal {H}^m(X_{k_i}(p,r/2)) \} \\&\le \liminf _{i\rightarrow \infty }\{\mathcal {F}^m(X_{k_i}{\setminus } A)\} - \frac{a \delta }{1+\delta }\, \mathbf {c} (r/2)^m \\&< \mathfrak {m}, \end{aligned}$$

a contradiction.

By (17),

$$\begin{aligned}&\mathcal {F}^m(X_k(p,s)) + \mathcal {F}^m((X_k\cap \mathcal {N}(p,r)){\setminus } B(p,s)) \le (1+\delta )\mathcal {F}^m(Z_k(p,s))\nonumber \\&\quad +(1+\delta )\mathcal {F}^m(Z_k{\setminus } B(p,s)), \end{aligned}$$

(19)

and thus

$$\begin{aligned}&\mathcal {F}^m(X_k(p,s)) \le (1+\delta )\mathcal {F}^m(Z_k(p,s)) + \delta \mathcal {F}^m(Z_k{\setminus } B(p,s)) \le (1+\delta )\mathcal {F}^m(Z_k(p,s))\nonumber \\&\quad + \,\delta \mathcal {F}^m(X_k{\setminus } A). \end{aligned}$$

(20)

$\square$

Let $K = 2b/a$ and $\epsilon _0 = 1/(2c_1mK)$. Let $\gamma _0$ denote the constant $``\gamma ''$ produced from Lemma 3.4 corresponding to n, C and $\epsilon _0$.

Lemma 4.7

Let $p \in X_0 {\setminus } A$ and $0<r\le d_p$. If $s\in D_p\cap (0,r)$, $\delta \le 1$, and $k \ge N_{p,r,\delta }$, then

$$\begin{aligned} \mathcal {H}^m(X_k(p,s)) \le K\gamma _0s^m + \frac{s}{2m} \frac{d}{ds}\mathcal {H}^m(X_k(p,s)) + \delta M/a. \end{aligned}$$

(21)

If in addition

$$\begin{aligned} \frac{\mathcal {H}^m(X_k(p,s))}{s^m} \ge 2K\gamma _0, \end{aligned}$$

(22)

then

$$\begin{aligned} \mathcal {H}^m(X_k(p,s)) \le \frac{s}{m} \frac{d}{ds}\mathcal {H}^m(X_k(p,s)) + 2\delta M/a. \end{aligned}$$

(23)

Proof

Let $\hat{X}_k$ denote the set $P_k \cup T_k \cup (X_k {\setminus } B(p,r))$, where $P_k$ and $T_k$ are the sets produced from Lemma 3.4 applied to $\text {``}X \text {''}=X_k$, $\text {``} r\text {''}=s$ and $\text {``}\epsilon \text {''}=\epsilon _0$. Lemmas 3.4, 4.6 and 4.5 then yield

$$\begin{aligned} \mathcal {H}^m(X_k(p,s))&\le K \mathcal {H}^m(\hat{X}_k(p,s)) + \delta M/a \end{aligned}$$

(24)

$$\begin{aligned}&\le K(\gamma _0 s^m + c_1 \epsilon _0 s \mathcal {H}^{m-1}(X_k\cap \partial B(p,s))) + \delta M/a\end{aligned}$$

(25)

$$\begin{aligned}&\le K\gamma _0s^m + \frac{s}{2m} \frac{d}{ds}\mathcal {H}^m(X_k(p,s)) + \delta M/a. \end{aligned}$$

(26)

The second assertion follows from algebraic manipulation of (21). $\square$

Let

$$\begin{aligned} c(p)=\max \{ M/(a d_p^m), 2K\gamma _0 \}. \end{aligned}$$

(27)

Theorem 4.8

Let $p \in X_0 {\setminus } A$ and $0<r< d_p$. Then

$$\begin{aligned} \limsup _{k \rightarrow \infty }\frac{\mathcal {H}^m(X_k(p,r))}{r^m}\le c(p). \end{aligned}$$

In particular,

$$\begin{aligned} \frac{\mu _0(B(p,r)) }{r^m}\le b c(p) \end{aligned}$$

(28)

Proof

Let $r<R<d_p$. By Lemmas 4.7 and 4.4, we may apply Lemma 3.6 to prove that

$$\begin{aligned} \limsup _{k \rightarrow \infty }\frac{\mathcal {H}^m(X_k(p,r))}{r^m}\le \max \left\{ \frac{M}{a R^m}, 2K\gamma _0 \right\} , \end{aligned}$$

using the inputs “$Y_k$” $= X_k$ and “$\eta$” $= \max \left\{ \frac{M}{a R^m}, 2K\gamma _0 \right\}$. Take $R\rightarrow d_p$.

Now (28) follows from the Portmanteau theorem, since $D_p$ is dense in $(0,d_p)$. $\square$

For $0<r<d_p$, let $c_r(p)=\sup _{ q\in X_0(p,r)} \{ c(q)\} <\infty$.

Corollary 4.9

If $p \in X_0{\setminus } A$ and $0<r<d_p$, then

$$\begin{aligned} 0< a\mathbf {c}/\alpha _m \le {\Theta _*}^m(\mu _0,q) \le {\Theta ^*}^m(\mu _0,q) \le bc_r(p)/\alpha _m < \infty \end{aligned}$$

for all $q \in X_0(p,r)$.

Proof

The lower bound is due to [22] Corollary 4.3.5. The upper bound follows from Theorem 4.8: We have by Lemma 4.4 and the Portmanteau theorem,

$$\begin{aligned} {\Theta ^*}^m(\mu _0,q)&= \limsup _{t \rightarrow 0, t\in D_q} \frac{\mu _0(B(q,t))}{\alpha _m t^m}\\&= \limsup _{t \rightarrow 0, t\in D_q} \lim _{k \rightarrow \infty } \frac{\mathcal {F}^m\lfloor _{X_k{\setminus } A}(B(q,t))}{\alpha _m t^m}\\&\le \limsup _{t \rightarrow 0, t\in D_q} \limsup _{k \rightarrow \infty } b\frac{\mathcal {H}^m(X_k(q,t)}{\alpha _m t^m}\\&\le \frac{b c(q)}{\alpha _m }\le \frac{b c_r(p)}{\alpha _m} < \infty . \end{aligned}$$

$\square$

Using [23] 6.9, we deduce the following corollary:

Corollary 4.10

If $p \in X_0{\setminus } A$ and $0<r<d_p$, then

$$\begin{aligned} a \mathbf {c} \mathcal {H}^m(X_0(p,r)) \le \alpha _m\mu _0(B(p,r)) \le b c_r(p) 2^m \mathcal {H}^m(X_0(p,r)). \end{aligned}$$

Corollary 4.11

$X_0{\setminus } \mathcal {N}_\epsilon (A)$ is semiregular (see footnote 5) for every $\epsilon >0$.

Proof

Let $Y=X_0{\setminus } \mathcal {N}_\epsilon (A)$. We show first that there exists a constant C such that if $x\in \mathbb {R}^n$ and $0<r\le R<\epsilon '\equiv \min {\mathbf {R}/4,\epsilon /4}$, then Y(x, R) can be covered by $C (R/r)^m$ balls of radius r. Indeed, suppose $\{p_i\}_{i\in I}$ is a maximal family of points in Y(x, R) that are of distance $\ge r$ from each other. Then, by (15) and Theorem 4.8,

$$\begin{aligned} r^m|I|= 2^m \sum _{i\in I}(r/2)^m&\le 2^m \sum _{i\in I} \frac{\mu _0(B(p_i,r/2))}{a\mathbf {c}}\\&\le 2^m \frac{\mu _0(\cup _{i\in I}B(p_i,r/2))}{a\mathbf {c}}\\&\le \frac{2^m}{a \mathbf {c}} \mu _0(B(x,2R))\\&\le \frac{2^m}{a \mathbf {c}} \mu _0(B(p,4R))\\&\le \frac{2^{3m}}{a \mathbf {c}} b c(p) R^m\\&\le \frac{2^{3m}}{a \mathbf {c}} b \max \{ M/(a (4\epsilon ')^m), 2K\gamma _0 \} R^m, \end{aligned}$$

where p is any point in $\{p_i\}_{i\in I}$. The last inequality is due to (27) and the fact that $4\epsilon '$ is a lower bound for $d_p$ for $p\in Y$.

The general case follows from the finiteness of $\mu _0$ and compactness of Y. Indeed, if $r<\epsilon '\le R$, then it is enough to show that Y can be covered by $C r^{-m}$ balls of radius r. The proof is the same as the first case, replacing $\mu _0(B(p,4R))$ with $\mu _0(\mathbb {R}^n)$ in the antepenultimate line. If $\epsilon '\le r \le R$, then it is enough to show that Y can be covered by C balls of radius $\epsilon '$, and such a finite C exists since Y is compact. $\square$

Corollary 4.12

If E is an approximate tangent m-plane for $X_0{\setminus } A$ at $p\in X_0{\setminus } A$, and $0<\epsilon <1$, then there exists $r>0$ such that $X_0(p,r){\setminus } \overline{\mathcal {C}(p,E,\epsilon )}=\varnothing$.

Proof

By (15) and Corollary 4.10, we may apply Lemma 3.7 to the set $X=X_0{\setminus } A$. $\square$

Theorem 4.13

(Rectifiability) The measure $\mu _0\lfloor _{\mathbb {R}^n{\setminus } A}$ is m-rectifiable.

Proof

By Corollary 4.10, it is enough to show that $X_0{\setminus } A$ is m-rectifiable. Write $X_0{\setminus } A=R\cup P$ where R is an m-rectifiable Borel set, P is purely m-unrectifiable and $R\cap P=\varnothing$ (see e.g. [23] 15.6.) If $\mathcal {H}^m(P)>0$, there exists by [23] 6.2 a point $p\in P$ such that $2^{-m}\le {\Theta ^*}^m(P,p)\le 1$ and ${\Theta ^*}^m(R,p)=0$. Thus, for every $\epsilon >0$ there exists $0<r<d_p/(2\sqrt{n})$ such that $\mathcal {H}^m(R(p,r\sqrt{n}))<\epsilon \,r^m$ and $\mathcal {H}^m(P(p,r)) > r^m \alpha _m 2^{-m-1}$. Let Q be a cube centered at p with side length 2r. Then $\mathcal {H}^m(P\cap Q)> diam (Q)^m 2^{-2m-1}\alpha _m$ and

$$\begin{aligned} \mathcal {H}^m(R\cap Q)<\epsilon \cdot diam (Q)^m. \end{aligned}$$

(29)

Since $\mathcal {H}^m(X_0{\setminus } A)<\infty$, there exists a cube $Q'$ disjoint from $\mathcal {N}(A,d_p/2)$ such that Q is contained in the interior of $Q'$,

$$\begin{aligned} \mathcal {H}^m(X_0\cap Q'{\setminus } Q)<\epsilon \cdot diam (Q')^m, \end{aligned}$$

(30)

$$\begin{aligned} \mathcal {H}^m(P\cap Q)> diam (Q')^m 2^{-2m-1} \alpha _m, \end{aligned}$$

(31)

and

$$\begin{aligned} \mu _0(\partial Q')=0. \end{aligned}$$

(32)

Now apply Lemma 3.5 to the cube $Q'$ and the sets $``U''=P\cap Q$ and $``E''=X_0\cap Q'$. By Corollary 4.11, the set E is semiregular, so the Lipschitz constant of the resulting map $\phi _1$ is bounded above by some constant $J<\infty$, independent of the choice of $\epsilon$, Q or $Q'$. Thus, by (30), (29), (3), and (31),

$$\begin{aligned} \mathcal {H}^m(\phi _1(X_0\cap Q'))&\le \mathcal {H}^m(\phi _1(X_0\cap Q'{\setminus } Q))+ \mathcal {H}^m(\phi _1(R\cap Q))+\mathcal {H}^m(\phi _1(P\cap Q)) \end{aligned}$$

(33)

$$\begin{aligned}&\le 2 J^m \epsilon \cdot diam (Q')^m\end{aligned}$$

(34)

$$\begin{aligned}&< \frac{\alpha _m}{2^{2m+2}} J^m \epsilon \,\mathcal {H}^m(P\cap Q)\end{aligned}$$

(35)

$$\begin{aligned}&\le \frac{\alpha _m}{2^{2m+2}} J^m \epsilon \,\mathcal {H}^m(X_0\cap Q'). \end{aligned}$$

(36)

We will need to apply the map $\phi _1$ to $X_k$ for k large. Since $X_k\rightarrow X_0$ in the Hausdorff metric, it follows from Lemma 3.1 (g) that there exists $N<\infty$ such that if $k>N$, then

$$\begin{aligned} \phi _1(X_k\cap Q')\subset S_{j,d}(Q')\cup \partial (Q'). \end{aligned}$$

(37)

For each $k\ge 1$ let $\nu _k$ denote the measure $\mathcal {H}^m\lfloor _{\phi _1(X_k\cap Q')}$ and let $\nu _0$ be a weak subsequential limit of $(\nu _k)_{k\in \mathbb {N}}$.

Note that ${supp}(\nu _0)\subset Q'$. More specifically, ${supp}(\nu _0)\subset \phi _1(X_0\cap Q')$, for if $x\notin \phi _1(X_0\cap Q')$ and $x\in Q'$, then since $\phi _1$ is proper, there is an open neighborhood V of x whose closure is disjoint from $\phi _1(X_0)$. Thus, for large enough k, we have $\phi _1(X_k)\cap \overline{V}=\varnothing$. So, $\nu _0(V)\le \limsup \nu _k(\overline{V})=0.$

Now suppose $x\in \phi _1(X_0\cap Q')$. By (37) it holds that for $s>0$ small enough,

$$\begin{aligned} \nu _0(B(x,s))&\le \limsup \nu _k(\mathcal {N}(x,2s))\\&\le \mathcal {H}^m(S_{j,d}(Q')\cup \partial (Q') \cap \mathcal {N}(x,2s))\\&\le {n \atopwithdelims ()m} \alpha _m (2s)^m. \end{aligned}$$

We conclude that ${\Theta ^*}^m(\nu _0,x)\le {n\atopwithdelims ()m}2^m$ for all $x\in {supp}(\nu _0)$ and so by [23] 6.9 and (36),

$$\begin{aligned} \nu _0(\mathbb {R}^n)\le {n\atopwithdelims ()m}2^{2m} \mathcal {H}^m(\phi _1(X_0\cap Q')) \le {n\atopwithdelims ()m} 2^{4m+2}\alpha _m^{-1} J^m \epsilon \,\mathcal {H}^m(X_0\cap Q'). \end{aligned}$$

Thus, there exists a sequence $k_i\rightarrow \infty$ with $\mathcal {H}^m(\phi _1(X_{k_i}\cap Q'))\le {n\atopwithdelims ()m} 2^{4m+3}\alpha _m^{-1} J^m \epsilon \,\mathcal {H}^m(X_0\cap Q'),$ and so

$$\begin{aligned} \mathcal {F}^m(\pi (\phi _1(X_{k_i}\cap Q'))) \le T \epsilon \,\mathcal {H}^m(X_0\cap Q'), \end{aligned}$$

(38)

where $T\equiv b \mathrm {Lip}(\pi )^m {n\atopwithdelims ()m} 2^{4m+3}\alpha _m^{-1} J^m.$ Therefore,

$$\begin{aligned} \mathcal {F}^m(\pi (\phi _1(X_{k_i})){\setminus } A) \le T \epsilon \,\mathcal {H}^m(X_0\cap Q')+ \mathcal {F}^m(X_{k_i}\cap (C{\setminus } Q'){\setminus } A). \end{aligned}$$

(39)

For large enough i, by Corollary 4.9, weak convergence, (32) and [23] 6.9,

$$\begin{aligned} diam (Q')^m \alpha _m/ 2^{2m+1} \le \mathcal {H}^m(X_0\cap Q') \le W \mathcal {F}^m(X_{k_i}\cap Q'), \end{aligned}$$

where $W=2\alpha _m/(a\mathbf {c}).$ Thus for $0< \epsilon < 1/(WT)$,

$$\begin{aligned} T \epsilon \,\mathcal {H}^m(X_0\cap Q') \le -(1/W - T \epsilon ) diam (Q')^m \alpha _m/ 2^{2m+1} + \mathcal {F}^m(X_{k_i}\cap Q'). \end{aligned}$$

Together with (39), this implies

$$\begin{aligned} \mathcal {F}^m(\pi (\phi _1(X_{k_i})){\setminus } A) \le \mathcal {F}^m(X_{k_i}{\setminus } A) -(1/W - T \epsilon ) diam (Q')^m \alpha _m/ 2^{2m+1} \end{aligned}$$

a contradiction for i large enough, since⁷ $\pi (\phi _1(X_{k_i}))^\dagger \in \mathcal {S}^m(C,A)$ and $\mathcal {F}^m(X_{k_i}{\setminus } A)\rightarrow \mathfrak {m}$.

$\square$

6 Lower semicontinuity

Given $0< \epsilon < 1$ and $p\in \mathbb {R}^n$, let $A(p,\epsilon ,r)$ denote the closed annular region $B(p,r) {\setminus } \mathcal {N}(p,(1-\epsilon )r)$. For $p\in C{\setminus } A$, let $C_{m,p} = 2c(p)m\frac{b}{a}$.

Lemma 5.1

Let $0< \epsilon < 1/2$ and $p\in X_0{\setminus } A$. There exist sequences of radii $r_i\in D_p$ and $N_i\in \mathbb {N}$ with $r_i \rightarrow 0$ such that

$$\begin{aligned} \mathcal {H}^m(X_k \cap A(p,\epsilon ,r_i)) \le C_{m,p}\epsilon r_i^m \end{aligned}$$

for all $k \ge N_i$.

Proof

If not, there exists $\delta > 0$ such that for every $r \in D_p\cap (0,\delta )$ there exists a subsequence $k_j\rightarrow \infty$ with

$$\begin{aligned} \mathcal {F}^m(X_{k_j}\cap A(p,\epsilon ,r)) > a C_{m,p}\epsilon r^m, \end{aligned}$$

(40)

and hence

$$\begin{aligned} \mu _0(A(p,\epsilon ,r)) \ge a C_{m,p}\epsilon r^m. \end{aligned}$$

(41)

Indeed, since $D_p$ is dense in $(0,d_p)$, (41) holds for all $r\in (0,\min \{d_p,\delta \})$.

Since $\mu _0$ is finite, there exist arbitrarily small $r\in (0,\min \{d_p,\delta \})$ such that $\mu _0(\partial A(p,\epsilon ,r_i))=0$ for all $i\ge 0$, where $r_i \equiv (1-\epsilon )^i r$. For such r, it follows from (41) that

$$\begin{aligned} \mu _0(B(p,r))&= \sum _{i=0}^\infty \mu _0(A(p,\epsilon ,r_i))\\&\ge a C_{m,p} \epsilon \sum _{i=0}^\infty (1-\epsilon )^{im} r^m\\&= a C_{m,p}\frac{\epsilon }{1-(1-\epsilon )^m} r^m\\&\ge 2c(p)b r^m, \end{aligned}$$

contradicting Theorem 4.8. $\square$

Theorem 5.2

If $V\subset C{\setminus } A$ is open, then

$$\begin{aligned} \mathcal {F}^m(X_0 \cap V) \le \liminf _{k\rightarrow \infty } \mathcal {F}^m(X_k \cap V). \end{aligned}$$

The idea of the proof of lower semicontinuity is to form a Vitali-type covering of $X_0 \cap V$ by disjoint balls $B(p_i,r_i)$ where $X_0$ has an approximate tangent plane $E_i$ at $p_i$. The ellipticity condition on f will provide a lower bound for $\mathcal {F}^m(X_k(p_i,r_i))$ in terms of $f(p_i,E_i) \alpha ^m r_i^m$ and a small error. Lusin’s theorem and rectifiability of $\mu _0$ will provide an upper bound for $F^m(X_0(p_i,r_i))$, again in terms of $f(p_i,E_i) \alpha ^m r_i^m$ and a small error.

Proof

Since $\mathcal {F}^m(X_k{\setminus } A)<\infty$, $k\ge 0$, it suffices to prove the claim for V disjoint from a neighborhood of A. Let $X_0'$ be the full $\mathcal {H}^m$ measure subset of $X_0\cap V$ consisting of those points p such that $\Theta ^m(X_0, p)=1$ and for which $X_0$ has an approximate tangent m-plane $E_p$ at p.

Fix $0<\epsilon <1/2$. By Lusin’s theorem there exists an $\mathcal {H}^m$ measurable $Z \subset X_0'$ for which the function $q \mapsto f(q,T_q X_0)$ is continuous and

$$\begin{aligned} \mathcal {F}^m(X_0' {\setminus } Z)\le \epsilon . \end{aligned}$$

(42)

Let $p \in Z$ and write $E\equiv p + E_p$. By Lemmas 3.7 and (15) there exists $0< d_p' < \min \{d_p, d_H(\{p\},C{\setminus } V)\}$ such that if $0< r < d_p'$, then

$$\begin{aligned} X_0(p,r)\subset \mathcal {N}(E, \epsilon r/2) \end{aligned}$$

(43)

and

$$\begin{aligned} (1-\epsilon ) \alpha _m r^m\le \mathcal {H}^m(X_0(p,r))\le (1+\epsilon ) \alpha _m r^m. \end{aligned}$$

(44)

By Lemma 5.1 there exist $N_j\in \mathbb {N}$ and $r_j\in D_p\cap (0,d_p')$ for each $j\in \mathbb {N}$, such that $r_j \rightarrow 0$ and

$$\begin{aligned} \mathcal {H}^m(X_k \cap A(p,\epsilon ,r_j)) \le C_V\epsilon r_j^m, \end{aligned}$$

(45)

for all $k \ge N_j$, where $C_V\equiv \sup _{p\in X_0\cap V} \{ C_{m,p}\}<\infty$.

An upper bound for $\mathcal {F}^m(X_0(p,r_j))$ By (44), for large enough j, we have

$$\begin{aligned} \mathcal {F}^m(Z(p,r_j))&= \int _{Z(p,r_j)} f(q,T_q X_0) d\mathcal {H}^m\\&\le (f(p,E_p) + \epsilon )(\mathcal {H}^m(X_0(p,r_j))\\&\le (f(p,E_p) + \epsilon )(\alpha _m r_j^m + \epsilon r_j^m)\\&\le f(p,E_p)\alpha _m r_j^m + K \epsilon r_j^m, \end{aligned}$$

where $K<\infty$ is independent of p, j and $\epsilon$. Thus,

$$\begin{aligned} \mathcal {F}^m(X_0(p,r_j)) \le f(p,E_p)\alpha _m r_j^m + K \epsilon r_j^m + \mathcal {F}^m((X_0'{\setminus } Z)(p,r_j)). \end{aligned}$$

(46)

A lower bound for $\mathcal {F}^m(X_k(p,r_j))$ Fix $j\in \mathbb {N}$. Since $X_k\rightarrow X_0$ in the Hausdorff metric, we may increase $N_j$ if necessary so that

$$\begin{aligned} X_k(p,r_j) \subset \mathcal {N}(E,\epsilon r_j/2) \end{aligned}$$

(47)

for all $k\ge N_j$.

For each $j\in \mathbb {N}$ there exists a Lipschitz map $\phi _j:\mathbb {R}^n\rightarrow \mathbb {R}^n$ such that

• (a) $\phi _j$ is the identity outside $A_j\equiv A(p,\epsilon ,r_j)\cap \mathcal {N}(E, \epsilon r_j)$;

• (b) $\phi _j(A_j)=A_j$;

• (c) On $A(p,\epsilon /3,(1-\epsilon /3)r_j)\cap \mathcal {N}(E, \epsilon r_j/2))$, the map $\phi _j$ is orthogonal projection onto E;

• (d) $\phi _j$ is $C^0$-close to a diffeomorphism;

• (e) The Lipschitz constant of $\phi _j$ depends only on n.

By Corollary 4.9 and (45) it holds that for small enough $\epsilon >0$, we may increase the constant $N_j$ so that if $k \ge N_j$, then $\phi _j(X_k)$ contains $E\cap \partial B(p, (1-\epsilon /2)r_j)$. Indeed, if $E\cap \partial B(p, (1-\epsilon /2)r_j)$ contains a point q that is not in $\phi _j(X_k)$, then we may orthogonally project $\phi _j(X_k(p,(1-\epsilon /2)r_j))$ onto E, and then radially project the resulting set away from q onto $E\cap \partial B(p, (1-\epsilon /2)r_j)$. For $\epsilon >0$ small enough, the image of $\phi _j(X_k)$ by this map will be contained in the neighborhood U of C, and we may apply⁸ the Lipschitz retraction $\pi :U \rightarrow C$ to create a new sequence $(Y_k)_{k\in \mathbb {N}}$ of elements of $\mathcal {S}^m(C,A)$. The sets $Y_k$ satisfy $Y_k=(X_k{\setminus } B(p,r_j))\cup Z_k$, where $\mathcal {H}^m(Z_k)\le \kappa ^m C_V \epsilon r_j^m$, and $\kappa <\infty$ depends only on the Lipschitz constants of $\phi _j$ and $\pi$. Using the density bounds in Corollary 4.9, we conclude that $\inf _k \{\mathcal {F}^m(Y_k)\}<\mathfrak {m}$ yielding a contradiction (c.f. the proof of Theorem 4.13 after (39).)

Moreover, there can be no retraction⁹ from $( \phi _j(X_k))(p, (1-\epsilon /2)r_j)$ onto $E\cap \partial B(p, (1-\epsilon /2)r_j)$, for if there exists such a retraction $\rho$, then by the Weierstrass theorem for locally compact spaces, it can be assumed without loss of generality to be Lipschitz. As a Lipschitz map, $\rho$ can then be extended by the identity to $\partial B(p, (1-\epsilon /2)r_j)$, then to the rest of $B(p, (1-\epsilon /2)r_j)$ using the Kirszbraun theorem, and then finally by the identity to all of $\mathbb {R}^n$. The sequence $(\pi (\rho (X_k)))_{k\in \mathbb {N}}$ will yield a contradiction for the same reason as above.

Thus, by the ellipticity of f and (45), it holds that for large enough j and $k\ge N_j$,

$$\begin{aligned} (f(p,E_p)-\epsilon )\alpha _m (1-\epsilon /2)^m r_j^m&\le \mathcal {F}_{\tilde{f}}^m(\phi _j(X_k)(p, (1-\epsilon /2)r_j))\\&\le \mathcal {F}^m(X_k(p, r_j)) + b' \mathrm {Lip}(\phi _j)C_V\epsilon r_j^m, \end{aligned}$$

or in other words,

$$\begin{aligned} f(p,E_p)\alpha _m r_j^m \le \mathcal {F}^m(X_k(p, r_j)) + K \epsilon r_j^m, \end{aligned}$$

(48)

where $K<\infty$ is independent of p, j and $\epsilon$.

A Covering By [23] Theorem 2.8, (46) and (48), there exists a covering $\{B(p_i,s_i)\}_{i\in I}$ of $\mathcal {H}^m$ almost all Z by disjoint closed balls $B(p_i,s_i)$ with $p_i \in Z$ and $0<s_i< d_p'$ small enough so that

$$\begin{aligned} \mathcal {F}^m(X_0(p_i,s_i)) \le f(p_i,E_{p_i})\alpha _m s_i^m + K \epsilon s_i^m + \mathcal {F}^m((X_0'{\setminus } Z)(p_i,s_i)) \end{aligned}$$

(49)

and

$$\begin{aligned} f(p_i,E_{p_i})\alpha _m s_i^m \le \mathcal {F}^m(X_k(p_i, s_i)) + K \epsilon s_i^m, \end{aligned}$$

(50)

for k large enough (depending on i.)

Choose a finite subcover $\{B(p_i,s_i)\}_{i=1}^N$ such that

$$\begin{aligned} \mathcal {F}^m(Z {\setminus } \cup _{i=1}^N Z(p_i,s_i)) < \epsilon . \end{aligned}$$

(51)

Associated to each $i=1,\dots ,n$ is the constant $N_i$ from (45). Let $N = \max _i \{N_i\}$. By (51), (49), (44), (50), and (42),

$$\begin{aligned} \mathcal {F}^m(X_0 \cap V)&\le 2\epsilon + \sum _{i=1}^N \mathcal {F}^m(X_0(p_i,s_i))\\&\le 2\epsilon + \sum _{i=1}^N f(p_i, E_{p_i})\alpha _m s_i^m + K \epsilon s_i^m + \mathcal {F}^m((X_0'{\setminus } Z)(p_i,s_i))\\&\le 3\epsilon + \sum _{i=1}^N \mathcal {F}^m(X_k(p_i, s_i)) + 2K \epsilon s_i^m\\&\le 3\epsilon + 4K \epsilon \mathcal {H}^m(X_0\cap V)/\alpha _m + \mathcal {F}^m(X_k \cap V), \end{aligned}$$

for all $k \ge N$. Therefore,

$$\begin{aligned} \mathcal {F}^m(X_0 \cap V) \le \liminf _{k \rightarrow \infty } \mathcal {F}^m(X_k\cap V) + C \epsilon , \end{aligned}$$

where $C<\infty$ is independent of $\epsilon$. Since this holds for all $\epsilon > 0$, the result follows. $\square$

In particular, $\mathcal {F}^m(X_0{\setminus } A) = \mathfrak {m}$. This completes the proof Theorem 2.5.