Abstract
We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed m dimensional subsets of \({\mathbf {R}}^n\) which is stable under taking smooth deformations homotopic to the identity and under local Hausdorff limits. We prove that the minimiser exists inside the class and is an \(({\mathscr {H}}^m,m)\) rectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a boundary. We admit unrectifiable and non-compact competitors and boundaries, and we make no restrictions on the dimension m and the co-dimension \(n-m\) other than \(1 \le m < n\). An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas follow Almgren’s (Ann Math (2) 87:321–391, 1968) paper. In the end we show that classes of sets spanning some closed set B in homological and cohomological sense satisfy our axioms.
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1 Introduction
The Plateau problem is about finding a minimiser of the area amongst the surfaces which span a given boundary. The notions of “area”, “surface”, and “spanning a boundary” of course need to be precisely defined so this problem actually has many different incarnations. Its history spans over two centuries and we have no intention of enumerating numerous prominent developments in this field. For the presentation of the classical formulations and solutions, their drawbacks, and the modern reformulations of the problem we refer the reader to the excellent expository articles by David [7] as well as by Harrison and Pugh [27]. These sources contain also an extensive list of references. We shall focus mainly on the comparison of our methods and results with the ones used in the papers published in the last years.
In this article we deal with an abstract Plateau’s problem which encompasses, e.g., the formulation of Adams and Reifenberg; cf. [37]. The notion of “area” of a competitor S is replaced by the value of a functional \(\Phi _F\) on S which is defined by integrating an elliptic integrand \(F : {\mathbf {R}}^n \times {\mathbf {G}}(n,m) \rightarrow [0,\infty ]\) with respect to the Hausdorff measure \({\mathscr {H}}^m\) over S—if S is \(({\mathscr {H}}^m,m)\) rectifiable, then we feed F with pairs (x, T), where \(x \in S\) and T is the approximate tangent plane to S at x. The integrand F provides an inhomogeneous (depending on the location) and anisotropic (depending on the tangent direction) weight for the Hausdorff measure. If \(F(x,T) = 1\) for all \((x,T) \in {\mathbf {R}}^n \times {\mathbf {G}}(n,m)\), then we call F the area integrand. Ellipticity means roughly that a flat m-dimensional disc D minimises \(\Phi _F\) amongst surfaces that cannot be retracted onto the boundary of D, i.e., \((m-1)\)-dimensional sphere; see 3.16. It can be seen as a geometric counterpart of quasi-convexity; see [32]. The “surfaces” and “boundaries” are, in our case, quite arbitrary closed subsets of \({\mathbf {R}}^n\)—not necessarily rectifiable nor compact. The notion of “spanning a boundary” does not appear at all. Our main Theorem (see 3.20) provides existence of an \(({\mathscr {H}}^m,m)\) rectifiable set which minimises \(\Phi _F\) (with F elliptic) inside an axiomatically defined class of competitors (see 3.4).
Section 3 contains all the definitions and the precise statement of the main theorem. In Sect. 12 we show that naturally defined (using Čech homology and cohomology) classes of sets spanning a given boundary (which may be an arbitrary closed set in \({\mathbf {R}}^n\)) satisfy our axioms.
Similar results were obtained recently by other authors. Harrison [23] suggested a new formulation of the problem, defined spanning employing the linking number, and used differential chains, developed earlier in [24], to find minimisers of the Hausdorff measure in co-dimension one. Harrison and Pugh [25, 26] proved existence of minimisers for the area integrand in arbitrary dimension and co-dimension using Čech cohomology to define spanning.
The same authors proved independently, in [28], a very similar result to ours. They showed existence of minimisers of an elliptic anisotropic and inhomogeneous functional in an abstractly defined class of competitors of any dimension and co-dimension. Their result admits non-Euclidean ambient metric spaces but, on the other hand, is restricted to the case when all competitors are compact and \(({\mathscr {H}}^m,m)\) rectifiable (however, Jenny Harrison informed the authors that the methods of [28] extend also to the case of non-compact competitors).
Even though, the main result of the current paper is so similar to [28], we emphasis that the method of the proof is different. In particular, we make extensive use of varifolds (especially, Almgren’s theory of slicing) and we provide a smooth deformation theorem.
De Lellis et al. [9] formulated the problem abstractly and showed existence of minimisers of the m-dimensional Hausdorff measure in any family of subsets of \({\mathbf {R}}^{m+1}\) containing enough competitors; see [9, Definition 1]. Later De Philippis et al. [10] generalised this result to any co-dimension assuming, roughly, that the set of competitors is closed under taking deformations which are uniform limits of maps \({\mathscr {C}}^1\) isotopic to identity; see [10, Definition 1.2]. After that, De Lellis et al. [8] obtained also minimisers for an inhomogeneous and anisotropic problem in co-dimension one. Finally, De Philippis et al. [11] also solved the problem in full generality.
These works all consider axiomatically defined families of competitors, which include, e.g., sets that span a boundary in the sense of Harrison [23] and sliding competitors of David [7]. However, in the latter case the results of [9, 10, 10, 11] do not ensure that the minimiser is a sliding deformation of the initial competitor.
The origin of our project was a mini-course that we conducted in 2014. We undertook the effort to understand Almgren’s existence result presented on the first 30 pages of [3]. Enlightened by his brilliant ideas we decided to present his approach to the Plateau problem in a sequence of lectures. The present manuscript was, at first, meant to serve as lecture notes for the mini-course but, with time, it grew into a full-fledged research paper containing new results.
The skeleton of the proof is the same as in [3] and our proofs of the intermediate steps are quite similar to Almgren’s but we also divert from [3] in many places. First of all we separated the abstract existence result from the application to a specific class of sets which homologically span a given boundary. Actually, in the definition of the good class of competitors, we do not use any notion of “spanning a boundary”—we only assume the class is closed under local Hausdorff convergence, and under taking images of sets with respect to certain admissible deformations; see 3.4. Moreover, we make no use of currents, flat chains, or G-varifolds in this paper. Second, we filled up most of the details that Almgren left to the reader. In particular, we had to develop a new smooth deformation theorem, which might be of separate interest (see 7.13) and we proved a perturbation Lemma (see 4.3) which allows to show rectifiability of minimisers. Third, we improved the main theorem by allowing for non-compact and unrectifiable competitors and boundaries.
Our deformation Theorem 7.13 takes some m dimensional sets \(\Sigma _1\), ..., \(\Sigma _l\) and a finite subfamily \({\mathcal {A}}\) of dyadic Whitney cubes and provides a \({\mathscr {C}}^{\infty }\) smooth homotopy \(f : [0,1] \times {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) between the identity and a map which deforms some neighbourhood G of \(\bigcup _{i=1}^l\Sigma _i \cap \bigcup {\mathcal {A}}\) onto an m dimensional skeleton of \({\mathcal {A}}\). Furthermore, for each \(t \in [0,1]\) the map \(f(t,\cdot )\) equals the identity outside some neighbourhood of \(\bigcup {\mathcal {A}}\). The main novelty with respect to well known constructions of this sort is that f is \({\mathscr {C}}^{\infty }\) smooth. This is important for two reasons. First, the push-forward by f defines a continuous map \(f_{\#} : {\mathbf {V}}_{m}({\mathbf {R}}^n) \rightarrow {\mathbf {V}}_{m}({\mathbf {R}}^n)\) on the space of m dimensional varifolds in \({\mathbf {R}}^n\). This allows to transfer certain estimates valid for the limit varifold (which, a priori, is not a competitor) onto elements of the minimising sequence; see, e.g., 9.1. Second, since the image of G under \(f(1,\cdot )\) is m dimensional, we may use a perturbation argument (see 4.3) to find another smooth map which is arbitrarily close to \(f(1,\cdot )\) in \({\mathscr {C}}^1\) topology and almost kills the measure of the unrectifiable part of \(\Sigma _i \cap \bigcup {\mathcal {A}}\). This allows to show that the minimiser coming from the main Theorem 3.20 is \(({\mathscr {H}}^m,m)\) rectifiable.
In contrast to the classical Federer–Fleming deformation theorem [19, 4.2.9] and Almgren’s deformation theorem [5, Chapter 1] ours works for quite arbitrary sets \(\Sigma \) in \({\mathbf {R}}^n\) which may not carry the structure of a rectifiable current. It differs also from a similar result of David and Semmes [14, Theorem 3.1] because we perform the deformation inside Whitney cubes of varying sizes and, in case \(\Sigma \) is \(({\mathscr {H}}^m,m)\) rectifiable, we provide estimates on the \({\mathscr {H}}^{m+1}\) measure of the whole deformation, i.e., on \({\mathscr {H}}^{m+1}(f [(0,1) \times \Sigma ])\). Moreover, our theorem is tailored especially for the use with varifolds which might not be rectifiable, so we actually prove estimates not on the \({\mathscr {H}}^m\) measure of \(f(1,\cdot )[\Sigma ]\) but rather on the integral \(\int _{\Sigma } \Vert \mathrm {D}f(1,\cdot )\Vert ^m \,\mathrm {d}{\mathscr {H}}^m\).
In the course of the proof of the main theorem we try to mimic, as much as possible, Almgren’s original ideas. In particular, rectifiability of the minimiser is proven employing the deformation theorem together with a perturbation argument based on the Besicovitch–Federer projection theorem; see Sect. 10. This point of the proof seems to make a lot of trouble in other works. De Lellis et al. [9] and De Philippis et al. [10] used the famous Preiss’ rectifiability Theorem [33]. De Lellis et al. [10] employed the theory of Caccioppoli sets, which is possible in co-dimension one. The first author in [17] used a very complex construction of Feuvrier [20] to modify a minimising sequence into a sequence consisting of quasi-minimal sets; cf. [4]. Harrison and Pugh [25, 28] choose a very special subsequence, which they call Reifenberg regular sequence, of the minimising sequence, whose elements enjoy good bounds on density ratios down to certain scale. We simply follow Almgren’s ideas.
In their final paper [11] De Philippis et al. make use of their very interesting result [12] yielding rectifiability for minima of certain elliptic functionals. Actually, in [12] they acquired a sufficient and necessary condition (called the atomic condition) on the integrand so that varifolds whose first variation with respect to such F induces a Radon measure are rectifiable. Later, De Rosa [13] showed that if F satisfies the atomic condition, then an F-minimising sequence of integral varifolds contains a sub-sequence converging to an integral varifold.
2 Notation
In principle we shall follow the notation of Federer; see [19, pp. 669–671]. However, we will use the standard abbreviations for intervals in \({\mathbf {R}}\), i.e., \((a,b) = \{ t \in {\mathbf {R}}: a< t < b\}\), \([a,b) = \{ t \in {\mathbf {R}}: a \le t < b\}\) etc. We reserve the symbol \(I = [0,1]\) for the closed unit interval. We will also write \(\{ x \in X : P(x) \}\) rather than \(X \cap \{ x : P(x) \}\) to denote the set of those \(x \in X\) which satisfy predicate P. For the identity map on some set X we use the symbol \(\mathrm {id}_{X} : X \rightarrow X\) and the characteristic function of X is denoted \(\mathbb {1}_X\) and is defined by \(\mathbb {1}_X(x) = 1\) if \(x \in X\) and \(\mathbb {1}_X(x) = 0\) if \(x \notin X\). If \(U \subseteq {\mathbf {R}}^m\) and \(V \subseteq {\mathbf {R}}^n\), the set of maps \(f : U \rightarrow V\) with continuous \(k^{\mathrm {th}}\) order derivatives is denoted by \({\mathscr {C}}^{k}(U,V)\). If \(f \in {\mathscr {C}}^k(U,V)\), we say that f is of class \({\mathscr {C}}^k\). In contrast to [19, 2.9.1] given two Radon measures \(\mu \) and \(\nu \) over \({\mathbf {R}}^n\) we write
Concerning varifolds we shall follow Allard’s notation; see [1]. In particular, if \(U \subset {\mathbf {R}}^n\) is open, we write \({\mathbf {V}}_{k}(U)\), \(\mathbf {IV}_{k}(U)\), and \(\mathbf {RV}_{k}(U)\) for the space of k dimensional varifolds, integral varifolds, and rectifiable varifolds in U following the definitions [1, 3.1, 3.5]. Also \({{\mathrm{VarTan}}}(V,x)\) shall denote the set of varifold tangents as defined in [1, 3.4]. In contrast to [1, 3.5], we shall write \({\mathbf {v}}_{k}(S)\) instead of \({\mathbf {v}}(S)\) to highlight the dimension of of the resulting varifold; cf. 3.10.
We recall some notation of Federer. As in [19, 2.2.6] we use the symbol \({\mathscr {P}}\) to denote the set of positive integers. The symbols \({\mathbf {U}}(a,r)\) and \({\mathbf {B}}(a,r)\) denote respectively the open and closed ball with centre a and radius r; see [19, 2.8.1]. We use the notation \(\varvec{\tau }_{a}\) and \(\varvec{\mu }_{s}\) for the translation by \(a \in {\mathbf {R}}^n\) and the homothety with ratio \(s \in {\mathbf {R}}\) respectively; see [19, 2.7.16, 4.2.8]. For the Hausdorff metric on compact subsets of \({\mathbf {R}}^n\) we write \({d_{{\mathscr {H}}}}\) and for the k dimensional Hausdorff measure \({\mathscr {H}}^k\); cf. [19, 2.10.21 and 2.10.2]. The scalar product of \(u,v \in {\mathbf {R}}^n\) is denoted \(u \bullet v\). The space of maps \(p \in {{\mathrm{Hom}}}({\mathbf {R}}^n,{\mathbf {R}}^m)\) such that \(p^* u \bullet p^* v = u \bullet v\) for all \(u,v \in {\mathbf {R}}^m\) (i.e. p is an orthogonal projection) is denoted \({\mathbf {O}}^*({n},{m})\); see [19, 1.7.4].
Following [3] and [6] if \(S \in {\mathbf {G}}(n,m)\) is an m dimensional linear subspace of \({\mathbf {R}}^n\), then \({S}_\natural \in {{\mathrm{Hom}}}({\mathbf {R}}^n,{\mathbf {R}}^n)\) shall denote the orthogonal projection onto S. In particular if \(p \in {\mathbf {O}}^*({n},{m})\) is such that \({{\mathrm{im}}}p^* = S\), then \({S}_\natural = p^* \circ p\).
Whenever \(\mu \) is a (Radon) measure over some set \(U \subseteq {\mathbf {R}}^n\) we sometimes use the same symbol \(\mu \) to denote the (not necessarily Radon) measure \(j_{\sharp }\mu \) over \({\mathbf {R}}^n\), where \(j : U \rightarrow {\mathbf {R}}^n\) is the inclusion map. Nonetheless, the support of \(\mu \) is always a subset of U, i.e., \({{\mathrm{spt}}}\mu \subseteq U\).
If A and B are subsets of some vectorspace, then we write \(A + B\) for the algebraic sum of A and B, i.e., the set \(\{ a + b : a \in A, b \in B\}\). If X and Y are vectorspaces, we write \(X \oplus Y\) for the the direct sum of X and Y; see [16, Chapter I, §2].
3 Statement of the main result
Definition 3.1
Let \(U \subseteq {\mathbf {R}}^n\) be open. We say that \(f : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) is a basic deformation in U if f is of class \({\mathscr {C}}^1\) and there exists a bounded convex open set \(V \subseteq U\) such that
If \(f \in {\mathscr {C}}^1({\mathbf {R}}^n,{\mathbf {R}}^n)\) is a composition of a finite number of basic deformations, then we say that f is an admissible deformation in U. The set of all such deformations shall be denoted \({\mathfrak {D}}({U})\).
Remark 3.2
In most cases the bounded convex set V shall be a cube or a ball.
Definition 3.3
Whenever \(K \subseteq {\mathbf {R}}^n\) is compact and \(A,B \subseteq {\mathbf {R}}^n\), we define \({d_{{\mathscr {H}},K}}(A,B)\) by
Definition 3.4
Let \(U \subseteq {\mathbf {R}}^n\) be an open set. We say that \({\mathcal {C}}\) is a good class in U if
-
(a)
\({\mathcal {C}} \ne \varnothing \);
-
(b)
each \(S \in {\mathcal {C}}\) is a closed subset of \({\mathbf {R}}^n\);
-
(c)
if \(S \in {\mathcal {C}}\) and \(f \in {\mathfrak {D}}({U})\), then \(f [S ]\in {\mathcal {C}}\);
-
(d)
if \(S_i \in {\mathcal {C}}\) for \(i \in {\mathscr {P}}\), and \(S \subseteq {\mathbf {R}}^n\), and \(\lim _{i \rightarrow \infty } {d_{{\mathscr {H}},K}}(S_i \cap U, S \cap U) = 0\) for all compact sets \(K \subseteq U\), then \(S \in {\mathcal {C}}\).
Remark 3.5
One example of a good class is given in 12.4. We expect that the methods presented in this article could work also if we assumed that admissible deformations are uniform limits of diffeomorphisms (so called monotone maps) as in [10, Definition 1.1]. In such case, the class denoted \({\mathcal {F}}(H,{\mathcal {C}})\) defined in [10, Definition 1.4] would also be good. However, we had trouble checking that all the deformations we use are monotone. In particular, the deformations constructed in 9.1 are clearly not monotone and there is no easy way to fix that. We anticipate that one could modify the deformation Theorem 7.13 to handle the situation from 9.1 and provide an appropriate monotone map but, given the overall complexity of the already presented material, we chose not to do that. A related result, allowing to approximate the cone construction by a sequence of diffeomorphisms and to get an isoperimetric inequality similar to (78), was obtained recently by Pugh [36].
Definition 3.6
(cf. [3, 1.2]) A function \(F : {\mathbf {R}}^n \times {\mathbf {G}}(n,m) \rightarrow [0,\infty )\) of class \({\mathscr {C}}^k\) for some non-negative integer \(k \in {\mathscr {P}}\) is called a \({\mathscr {C}}^k\) integrand.
if, additionally, \(\inf {{\mathrm{im}}}F / \sup {{\mathrm{im}}}F \in (0,\infty )\), then we say that F is bounded.
Definition 3.7
(cf. [3, 3.1]) If \(\varphi \in {\mathscr {C}}^{1}({\mathbf {R}}^n,{\mathbf {R}}^n)\) and F is an integrand, then the pull-back integrand \(\varphi ^\#F\) is given by by
If \(\varphi \) is a diffeomorphism, then the push-forward integrand is given by \(\varphi _\#F = (\varphi ^{-1})^\#F\).
Definition 3.8
(cf. [3, 1.2]) If F is a \({\mathscr {C}}^k\) integrand and \(x \in {\mathbf {R}}^n\), then we define another \({\mathscr {C}}^k\) integrand \(F^x\) by the formula
Remark 3.9
Recall that \(\varvec{\gamma }_{n,m}\) denotes the canonical probability measure on \({\mathbf {G}}(n,m)\) invariant under the action of the orthogonal group \({\mathbf {O}}({n})\); see [19, 2.7.16(6)].
Definition 3.10
(cf. [1, 3.5]) Assume \(S \subseteq {\mathbf {R}}^n\) is \({\mathscr {H}}^m\) measurable and such that \({\mathscr {H}}^m(S \cap K) < \infty \) for any compact set \(K \subseteq {\mathbf {R}}^n\). We define \({\mathbf {v}}_{m}(S) \in {\mathbf {V}}_{m}({\mathbf {R}}^n)\) in the following way: first decompose S into a sum \(S_u \cup S_r\), where \(S_u\) is purely \(({\mathscr {H}}^m,m)\) unrectifiable and \(S_r\) is countably \(({\mathscr {H}}^m,m)\) rectifiable and Borel (cf. [19, 3.2.14]); then set
Definition 3.11
If F is a \({\mathscr {C}}^k\) integrand, we define the functional \(\Phi _F : {\mathbf {V}}_{m}({\mathbf {R}}^n) \rightarrow [0,\infty ]\) by the formula
Remark 3.12
If \({{\mathrm{spt}}}\Vert V\Vert \) is compact we have \(\Phi _F(V) = V(\gamma F)\), whenever \(\gamma \in {\mathscr {D}}({\mathbf {R}}^n,{\mathbf {R}})\) is such that \({{\mathrm{spt}}}\Vert V\Vert \subseteq \gamma ^{-1} \{1\}\).
Definition 3.13
If \(S \subseteq {\mathbf {R}}^n\) is \({\mathscr {H}}^m\) measurable, satisfies \({\mathscr {H}}^m(S \cap K) < \infty \) for all compact sets \(K \subseteq {\mathbf {R}}^n\), and \(S_u \subseteq S\) is a purely \(({\mathscr {H}}^m,m)\) unrectifiable part of S, then we set
For any other subset S of \({\mathbf {R}}^n\) we set \(\Psi _F(S) = \Phi _F(S) = \infty \).
Remark 3.14
Assume \(V \in {\mathbf {V}}_{m}({\mathbf {R}}^n)\), \(\varphi : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) is of class \({\mathscr {C}}^1\), and F is a \({\mathscr {C}}^0\) integrand. Then
If \(S \subseteq {\mathbf {R}}^n\) is \({\mathscr {H}}^m\) measurable and satisfies \({\mathscr {H}}^m(S \cap K) < \infty \) for all compact sets \(K \subseteq {\mathbf {R}}^n\), then
given S is countably \(({\mathscr {H}}^m,m)\) rectifiable or \(\varphi = \varvec{\mu }_{r}\) for some \(r \in (0,\infty )\) or \(\varphi = \varvec{\tau }_{a}\) for some \(a \in {\mathbf {R}}^n\).
Remark 3.15
If S is \({\mathscr {H}}^m\) measurable, \({\mathscr {H}}^m(S \cap K) < \infty \) for any compact \(K \subseteq {\mathbf {R}}^n\), \(S = S_u \cup S_r\), where \(S_u\) is purely \(({\mathscr {H}}^m,m)\) unrectifiable and \(S_r\) is countably \(({\mathscr {H}}^m,m)\) rectifiable, F is an integrand, \(x \in {\mathbf {R}}^n\), then
We shall use the following notion of ellipticity based on the definition given by Almgren; see [4, IV.1(7)] and [3, 1.6(2)]. It can be understood as a geometric version of quasi-convexity; cf. [32]. Similar notion for currents can be found in [19, 5.1.2].
Definition 3.16
An \({\mathscr {C}}^0\) integrand F is called elliptic if there exists a continuous function \(c : {\mathbf {R}}^n \rightarrow (0,\infty )\) such that for all \(T \in {\mathbf {G}}(n,m)\) we have
whenever
-
(a)
\(D = {\mathbf {B}}(0,1) \cap T\) is a unit m dimensional disc in T;
-
(b)
S is a compact subset of \({\mathbf {R}}^n\), \({\mathscr {H}}^m(S) < \infty \), and S cannot be deformed onto \(R = {{\mathrm{Bdry}}}{\mathbf {B}}(0,1) \cap T\) by any Lipschitz map \(f : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) satisfying \(f(x) = x\) for \(x \in R\).
Remark 3.17
Note the following observations.
-
The area integrand \(F \equiv 1\) is elliptic.
-
If F is an integrand and \(\varphi \in {\mathscr {C}}^{1}({\mathbf {R}}^n,{\mathbf {R}}^n)\) is a diffeomorphism, then F is elliptic if and only if \(\varphi _\#F\) is elliptic; cf. [3, 4.3].
-
Any convex combination of elliptic integrands is elliptic.
Remark 3.18
In the original definition of Almgren one assumes also that S is \(({\mathscr {H}}^m,m)\) rectifiable. Since we want to work with possibly unrectifiable competitors we need to drop this assumption. For the same reason we used \(\Psi _{F^x}\) instead of \(\Phi _{F^x}\) in 3.16.
Assume \(x \in {\mathbf {R}}^n\) is fixed and \(T \in {\mathbf {G}}(n,m)\) is such that \(F(x,T) = M = \sup \{ F(x,P) : P \in {\mathbf {G}}(n,m)\}\). Set \(E = \int F(x,P) \,\mathrm {d}\varvec{\gamma }_{n,m}(P)\), \(D = {\mathbf {B}}(0,1) \cap T\), and \(R = {{\mathrm{Bdry}}}{\mathbf {B}}(0,1) \cap T\). Assume \(E < M\) and \(S = D {{\mathrm{\sim }}}F \cup W\) satisfies 3.16(b), where \(F \subseteq D\) is closed and W is purely \(({\mathscr {H}}^m,m)\) unrectifiable and closed, and \(W \cap (D {{\mathrm{\sim }}}F) = \varnothing \). Then
If one could perform this construction ensuring \({\mathscr {H}}^m(W) < M/E {\mathscr {H}}^m(F)\), then the above quantity would become negative. Hence, assuming such construction is possible, if we used \(\Phi _{F^x}\) in place of \(\Psi _{F^x}\) in 3.16, then there would be no elliptic integrands depending non-trivially on the second variable. In some directions, filling a hole with purely unrectifiable set would always be better then filling it with a flat disc. Nonetheless, we believe this is not possible.
In a forthcoming article, the second author and Antonio De Rosa show that one obtains an equivalent definition of ellipticity if one assumes in 3.16(b) that S is \(({\mathscr {H}}^m,m)\) rectifiable.
Remark 3.19
It is not clear whether strict convexity of F in the second variable is enough to ensure ellipticity of F as is the case in the context of currents; see [19, 5.1.2]. De Lellis et al. where able to prove their existence result assuming F is uniformly convex in the second variable and of class \({\mathscr {C}}^2\); see [10, Definition 2.4]. De Philippis et al. defined the so called atomic condition for the integrand; see [12, Definition 1]. In co-dimension one this condition is equivalent to strict convexity of F in the second variable; see [12, Theorem 1.3]. Moreover, varifolds whose first variation with respect to F, satisfying the atomic condition, induces a Radon measure are rectifiable; see [12, Theorem 1.2].
Our main theorem reads.
Theorem 3.20
Let \(U \subset {\mathbf {R}}^n\) be an open set, \({\mathcal {C}}\) be a good class in U, and F be a bounded elliptic \({\mathscr {C}}^0\) integrand. Set \(\mu = \inf \bigl \{ \Phi _F(T \cap U) : T \in {\mathcal {C}} \bigr \}\).
If \(\mu \in (0,\infty )\), then there exist \(S \in {\mathcal {C}}\) and a sequence \(\{ S_i \in {\mathcal {C}} : i \in {\mathscr {P}}\}\) such that
-
(a)
\(S \cap U\) is \(({\mathscr {H}}^m,m)\) rectifiable. In particular \({\mathscr {H}}^m(S \cap U) < \infty \).
-
(b)
\(\lim _{i \rightarrow \infty } \Phi _F(S_i \cap U) = \Phi _F(S \cap U) = \mu \).
-
(c)
\(\lim _{i \rightarrow \infty } {\mathbf {v}}_{m}(S_i \cap U) = {\mathbf {v}}_{m}(S \cap U)\) in \({\mathbf {V}}_{m}(U)\).
-
(d)
\(\lim _{i \rightarrow \infty } {d_{{\mathscr {H}},K}}(S_i \cap U, S \cap U) = 0\) for any compact set \(K \subseteq U\).
Furthermore, if \({\mathbf {R}}^n {{\mathrm{\sim }}}U\) is compact and there exists a \(\Phi _F\)-minimising sequence in \({\mathcal {C}}\) consisting only of compact sets (but not necessarily uniformly bounded), then
4 Unrectifiable sets under submersions
Assume \(U \subseteq {\mathbf {R}}^n\) is open and \(K \subseteq U\) is purely \(({\mathscr {H}}^m,m)\) unrectifiable with \({\mathscr {H}}^m(K) < \infty \) and \(f \in {\mathscr {C}}^{1}({\mathbf {R}}^n, {\mathbf {R}}^n)\) is such that \(\mathrm {D}f(x)\) is of rank at most m for \(x \in U\). We construct an arbitrarily small \({\mathscr {C}}^1\) perturbation \({{\tilde{f}}}\) of f such that \({\mathscr {H}}^m({{\tilde{f}}}[K ])\) becomes very small. Additionally, we ensure that \({{\tilde{f}}}\) is of the form \({{\tilde{f}}} = f \circ \rho \), where \(\rho \) is a diffeomorphism of \({\mathbf {R}}^n\) such that \({{\mathrm{Lip}}}(\rho - \mathrm {id}_{{\mathbf {R}}^n})\) is very small and \(\rho [ U ] = U\). This provides a useful feature of \({{\tilde{f}}}\), namely that \(\tilde{f} [ U ] \subseteq f [ U ]\).
A similar result was proven recently by Pugh [35]. It could be possible to obtain \({\mathscr {H}}^m({{\tilde{f}}}[K ]) = 0\) as was shown by Gałęski [22] but for our purposes it suffices only to make the measure small. Also the map constructed in [22] kills only the measure of the part of K on which \(\dim {{\mathrm{im}}}\mathrm {D}f(x) = m\) and we need to take care also of the part where the rank of \(\mathrm {D}f\) is strictly less than m. Finally, we should mention that Almgren alluded that such result should hold already in [3, 2.9(b1)].
In the next preparatory lemma we construct a smooth map \(M : {\mathbf {R}}\rightarrow {\mathbf {O}}({n})\) which continuously rotates a given m-plane S onto another given m-plane T. We also derive estimates on \(M'\) as well as on \(\Vert M(\cdot ) - \mathrm {id}_{{\mathbf {R}}^n}\Vert \) in terms of \(\Vert {S}_\natural - {T}_\natural \Vert \).
Lemma 4.1
Let n and m be positive integers such that \(0 < m \le n\). There exists \(\Gamma \in (0,\infty )\) such that for all \(S,T \in {\mathbf {G}}(n,m)\) there exists \(M : {\mathbf {R}}\rightarrow {{\mathrm{Hom}}}({\mathbf {R}}^n,{\mathbf {R}}^n)\) of class \({\mathscr {C}}^\infty \) satisfying
Proof
We shall construct the map M similarly as in [1, 8.9(3)]. First set
Observe that A, B, C, D are pairwise orthogonal and sum up to the whole of \({\mathbf {R}}^n\), i.e.,
Note also that there exist natural numbers k and l such that
For our convenience we set \(S_0 = S\) and \(T_0 = T\). If \(k > 0\), then we shall construct inductively
-
subspaces \(S \supseteq S_1 \supseteq S_2 \supseteq \cdots \supseteq S_k\) and \(T \supseteq T_1 \supseteq T_2 \supseteq \cdots \supseteq T_k\) and \(V_1, \ldots , V_k \subseteq S + T\),
-
and vectors \(s_1 \in S_1\), ..., \(s_k \in S_k\) and \(t_1 \in T_1\), ..., \(t_k \in T_k\).
To start the construction we set \(S_1 = S \cap D\) and \(T_1 = T \cap D\). Then we use [1, 8,9(3)] to find \(s_1 \in S_1\) so that \(|s_1| = 1\) and \(|{(T^{\perp }_1)}_\natural s_1| = \Vert {S_1}_\natural - {T_1}_\natural \Vert \). Note that \(\Vert {S_1}_\natural - {T_1}_\natural \Vert < 1\) because the spaces \(S_1\) and \(T_1\) are orthogonal to C. Hence, we may define \(t_1 = {T_1}_\natural s_1 |{T_1}_\natural s_1|^{-1}\) and \(V_1 = {{\mathrm{span}}}\{ s_1, t_1 \}\). Assuming we have constructed \(S_1\), ..., \(S_i\) and \(T_1\), ..., \(T_i\) and \(s_1\), ..., \(s_i\) and \(t_1\), ..., \(t_i\) for some \(i \in \{1,\ldots ,k-1\}\) we proceed by requiring
Observe that
thus,
Therefore,
Clearly \((s_1,\ldots ,s_k)\) and \((t_1,\ldots ,t_k)\) are orthonormal bases of \(S \cap D\) and \(T \cap D\) respectively. Next, we choose arbitrary orthonormal bases \((s_{k+1},\ldots ,s_{k+l})\) of \(S \cap T^{\perp }\) and \((t_{k+1},\ldots ,t_{k+l})\) of \(T \cap S^{\perp }\) and \((e_1,\ldots ,e_{n-2(k+l)})\) of \(A \oplus B\). We also define
and note that \(0 < \alpha _i \le \pi /2\) by construction. Now we are in position to define M. It shall be the identity on \(A \oplus B\) and on each \(V_i = {{\mathrm{span}}}\{s_i,t_i\}\) it will be the rotation sending \(s_i\) to \(t_i\) for \(i = 1,2,\ldots ,k+l\). More precisely we set
and define for \(\tau \in {\mathbf {R}}\)
Since \(\{s_1,\ldots ,s_{k+l},{\hat{s}}_1,\ldots ,{\hat{s}}_{k+l},e_1,\ldots ,e_{n-2(k+l)}\}\) is an orthonormal basis of \({\mathbf {R}}^n\) we see that \(M(\tau ) \in {\mathbf {O}}({n})\) for each \(\tau \in {\mathbf {R}}\). It is also immediate that \(M(0) = \mathrm {id}_{{\mathbf {R}}^n}\) and \(M(1) [S ]= T\).
To prove (2) we first estimate \(\alpha _i\). Recall that \(1 - \cos x = 2 \sin ^2(x/2)\) for \(x \in {\mathbf {R}}\) and \(|x| \le 2 |\sin x|\) whenever \(|x| \le \pi /2\); hence, for \(i = 1,\ldots ,k+l\)
If \(\Vert {T}_\natural - {S}_\natural \Vert < 1/2\), then we use standard estimates \(\exp (x) \ge 1 + x\) and \(\log (1+x) \ge x/(1+x)\) valid for \(x > -1\) to derive
If \(\Vert {T}_\natural - {S}_\natural \Vert \ge 1/2\), we have \(\bigl (1 - \bigl (1 - \Vert {T}_\natural - {S}_\natural \Vert ^2\bigr )^{1/2}\bigr )^{1/2} \le 1 \le 2 \Vert {T}_\natural - {S}_\natural \Vert \) so (3) holds also in this case.
Using \(|\sin x|\le |x|\) for \(x \in {\mathbf {R}}\) and (3) we obtain for \(i = 1,2,\ldots ,k+l\) and \(\tau \in {\mathbf {R}}\)
Thus, whenever \(v \in {\mathbf {R}}^n\) and \(|v| = 1\),
which proves the first part of (2). By direct computation we obtain \(|M'(\tau ) s_i| = |M'(\tau ) {\hat{s}}_i| = \alpha _i\) for \(\tau \in {\mathbf {R}}\) and \(i = 1,2,\ldots ,k+l\). Therefore, employing (3),
\(\square \)
The following technical lemma is a localised and reparameterised version of 4.1. Roughly speaking, we construct a diffeomorphism \(\rho \) of \({\mathbf {R}}^n\) which acts as a rotation inside a given ball and is the identity outside some neighbourhood of that ball. To be able to utilise 4.2 in 4.3 we need to perform the rotation in different coordinates, which is accomplished by passing through a diffeomorphism \(\varphi \). We use estimates from 4.1 to bound \({{\mathrm{Lip}}}(\rho - \mathrm {id}_{{\mathbf {R}}^n})\).
Lemma 4.2
Assume
Then there exist a diffeomorphism \(\rho \in {\mathscr {C}}^k({\mathbf {R}}^n,{\mathbf {R}}^n)\) and \(p \in {\mathbf {O}}^*({n},{m})\) such that
Proof
Employ 4.1 to obtain a smooth map \(M : {\mathbf {R}}\rightarrow {{\mathrm{Hom}}}({\mathbf {R}}^n,{\mathbf {R}}^n)\) such that \(M(1)[S ]= T\) and \(M(\tau ) \in {\mathbf {O}}({n})\) for each \(\tau \in {\mathbf {R}}\). Let \(\zeta : {\mathbf {R}}\rightarrow {\mathbf {R}}\) be of class \({\mathscr {C}}^\infty \) and satisfy \(\zeta (t) = 0\) for \(t \le 0\), and \(\zeta (t) = 1\) for \(t \ge 1\), and \(0 \le \zeta '(t) \le 2\) for \(t \in {\mathbf {R}}\), and \(0 \in {{\mathrm{Int}}}\zeta ^{-1} \{0\}\), and \(1 \in {{\mathrm{Int}}}\zeta ^{-1} \{1\}\). Define \(\pi : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\), and \(\eta : {\mathbf {R}}^n \rightarrow {\mathbf {R}}\), and \(p \in {{\mathrm{Hom}}}({\mathbf {R}}^n,{\mathbf {R}}^m)\) by requiring
Note that \(\eta \) is Lipschitz continuous and \(\pi \) is of class \({\mathscr {C}}^k\) because \(0 \in {{\mathrm{Int}}}\zeta ^{-1} \{0\}\) and \(1 \in {{\mathrm{Int}}}\zeta ^{-1} \{1\}\). Moreover, \(p \in {\mathbf {O}}^*({n},{m})\) and
Hence, we can set
Clearly (8), (9), (10) imply (5) and (6) and we only need to check (7). Recalling (2) and (4) and \({{\mathrm{Lip}}}(\varphi |_{{\mathbf {B}}(a,r)}) \le L\) we estimate for \(x \in {\mathbf {B}}(a,r)\)
In particular, using (9) we conclude that \({{\mathrm{Lip}}}(\pi - \mathrm {id}_{{\mathbf {R}}^n}) < 1\), so \(\pi \) and \(\rho \) are diffeomorphisms. Employing (11) we see also that if \(x \in {\mathbf {B}}(a,r)\), then
where \(y \in U {{\mathrm{\sim }}}{\mathbf {U}}(a,r)\) is any point such that \(|x-y| = {{\mathrm{dist}}}(x, {\mathbf {R}}^n {{\mathrm{\sim }}}{\mathbf {B}}(a,r)) \le r\). Utilising (11) and (12) we see that \({{\mathrm{Lip}}}(\pi ) \le 2\) and for \(x \in {\mathbf {B}}(a,r)\)
\(\square \)
In the next lemma given an open set \(U \subseteq {\mathbf {R}}^n\), a purely \(({\mathscr {H}}^m,m)\) unrectifiable set \(K \subseteq U\) with \({\mathscr {H}}^m(K) < \infty \) and a map \(f \in {\mathscr {C}}^{k}({\mathbf {R}}^n, {\mathbf {R}}^n)\) such that \(\mathrm {D}f(x)\) is of rank at most m for \(x \in U\) we employ the constant rank theorem together with a Vitali covering theorem to get a family of balls in each of which we apply 4.2 and the Besicovitch–Federer projection theorem to construct a diffeomorphism \(\rho \) of \({\mathbf {R}}^n\) such that \(f \circ \rho [K ]\) has significantly less \({\mathscr {H}}^m\) measure than K itself. Since, in general, f may map the set where \(\mathrm {D}f(x)\) has rank strictly less than m into a set of positive \({\mathscr {H}}^m\) measure we need to additionally assume that this does not happen or assume \(k \ge n - m + 1\) and employ the Morse–Sard theorem; see 4.4.
Lemma 4.3
Let \(K \subseteq {\mathbf {R}}^n\) be purely \(({\mathscr {H}}^m,m)\) unrectifiable with \({\mathscr {H}}^m(K) < \infty \). Let \(f : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) be of class \({\mathscr {C}}^{k}\) with \(k \ge 1\). Suppose there exists an open set \(U \subseteq {\mathbf {R}}^n\) such that
Then for any \(\varepsilon \in (0,\infty )\) there exists a diffeomorphism \(\rho _\varepsilon : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) of class \({\mathscr {C}}^{k}\) such that
Proof
Let \(\varepsilon \in (0,\infty )\) and let \(q : {\mathbf {R}}^m \times {\mathbf {R}}^{n - m} \rightarrow {\mathbf {R}}^m\) be given by \(q(x,y) = x\). Set
Since \(\dim {{\mathrm{im}}}\mathrm {D}f(x) \le m\) for all \(x \in U\) we see that \(A = \{ x \in U : \bigwedge _{m} \mathrm {D}f(x) \ne 0 \}\) is open. Hence, for every \(a \in A\) the constant rank theorem [19, 3.1.18] ensures the existence of open sets \(U_a \subseteq U\), \(V_a \subseteq {\mathbf {R}}^n\), maps \(\varphi _a : U_a \rightarrow {\mathbf {R}}^n\), \(\psi _a : V_a \rightarrow {\mathbf {R}}^n\) which are diffeomorphisms onto their respective images, and orthogonal projections \(p_a \in {\mathbf {O}}^*({n},{m})\) such that
Applying the Vitali covering theorem (see [19, 2.8.16, 2.8.18] or alternatively [30, 2.8]) to the measure 
and the family of all the closed balls \({\mathbf {B}}(a,r)\) satisfying
we obtain a countable disjointed collection \({\mathcal {B}}\) of closed balls having the properties (14), (15), and additionally
Whenever \({\mathbf {B}}(a,r) \in {\mathcal {B}}\) we set
Set \(I = \{ a \in {\mathbf {R}}^n : {\mathbf {B}}(a,r) \in {\mathcal {B}} \text { for some } r \in {\mathbf {R}}\}\) and \(T = {{\mathrm{im}}}(q^*) = {\mathbf {R}}^m \times \{0\} \in {\mathbf {G}}(n,m)\). Whenever \(a \in I\) define \(r_a \in {\mathbf {R}}\) to be the unique number such that \({\mathbf {B}}(a,r_a) \in {\mathcal {B}}\). Suppose \(a \in I\). Since \(\varphi _a\) is a diffeomorphism onto its image, we see that \(\varphi _a[K \cap {\mathbf {B}}(a,r_a) ]\) is purely \(({\mathscr {H}}^m,m)\) unrectifiable. Hence, the Besicovitch–Federer projection theorem (see [19, 3.3.15] or alternatively [30, 18.1]) allows us to find a sequence of m-planes \(S_{a,i} \in {\mathbf {G}}(n,m)\) such that \(\Vert {S_{a,i}}_\natural - {T}_\natural \Vert \rightarrow 0\) as \(i \rightarrow \infty \) and
Using (15) we find \({\tilde{r}}_a \in {\mathbf {R}}\) such that \(0< {\tilde{r}}_a < r_a\) and
Set \(\Delta = \Gamma _{{4.2}}(L_a, r_a, {\tilde{r}}_a)\) and
Choose \(i_a \in {\mathscr {P}}\) so big that
Employ 4.2 with \(S_{a,i_a}\), \(\varphi _a\), \(r_a\), \({\tilde{r}}_a\) in place of S, \(\varphi \), r, \({{\tilde{r}}}\) to obtain a diffeomorphism \(\rho = \rho _{a} \in {\mathscr {C}}^k({\mathbf {R}}^n,{\mathbf {R}}^n)\) and a projection \(p = p_{a} \in {\mathbf {O}}^*({n},{m})\) satisfying (5), (6), (7).
To finish the construction, we set
Since \({\mathcal {B}}\) is disjointed and each \(\rho _a\) is the identity outside the corresponding ball \({\mathbf {B}}(a,r_a) \in {\mathcal {B}}\), we see that \(\rho _{\varepsilon }\) is a well defined diffeomorphism of class \({\mathscr {C}}^k\). Moreover, using (13) and (16), then (6) and (5) together with (17) and finally (7) combined with (18) we obtain
Recalling (7) and (19) we see also
and
\(\square \)
Remark 4.4
If \(k \ge n - m + 1\), then the Morse–Sard theorem [19, 3.4.3] implies that \({\mathscr {H}}^m(f[\{ x \in K : \dim {{\mathrm{im}}}\mathrm {D}f(x) < m \} ]) = 0\) and assumption (13) becomes redundant.
Corollary 4.5
Set \(g = f \circ \rho _{\varepsilon }\) and
Then for \(x \in {\mathbf {R}}^n\) we obtain
In particular, for \(x \in {\mathbf {R}}^n\)
5 Smooth almost retraction of \({\mathbf {R}}^n\) onto a cube
In this section we construct, in 5.3, a \({\mathscr {C}}^{\infty }\) function which maps all of \({\mathbf {R}}^n\) onto the cube \(Q = [-\,1,1]^n\). This mapping is not a retraction because it moves points inside the cube Q. Its main features are that it is smooth and it preserves all the lower dimensional skeletons of Q and even the skeletons of the neighbouring dyadic cubes of side length 1. As a corollary of 5.3 we produce, in 5.4, a function which maps a small neighbourhood of Q onto Q and is the identity outside a bit larger neighbourhood of Q. We also carefully track the Lipschitz constants of the mappings.
First we need to introduce some notation to be able to conveniently handle various faces of the cube \([-\,1,1]^n\) and its dyadic neighbours.
Remark 5.1
Let \(e_1\), ..., \(e_n\) be the standard basis of \({\mathbf {R}}^n\). Set \(Q = \{ x \in {\mathbf {R}}^n : x \bullet e_j \le 1 \text { for }\; j = 1,2,\ldots ,n \} = [-\,1,1]^n\). For \(\kappa = (\kappa _1, \ldots , \kappa _n) \in \{-\,1,0,1\}^n\) define
Observe that the sets \(C_{\kappa }\) for \(\kappa \in \{-\,1,0,1\}^n\) are convex, have nonempty interiors, are pairwise disjoint, and form a partition of \({\mathbf {R}}^n\), i.e.,
For \(\kappa \in \{-\,1,0,1\}^n\) we have \(\dim (T_{\kappa }) = {\mathscr {H}}^0(\{ j : \kappa _j = 0 \})\), and \(F_{\kappa }\) is a \(\dim (T_{\kappa })\) dimensional face of Q lying in the affine space \(c_{\kappa } + T_{\kappa }\), and \(F_{\kappa }\) is relatively open in \(c_{\kappa } + T_{\kappa }\), and \(c_{\kappa }\) is the centre of \(F_{\kappa }\). In particular, \(C_{(0,0,\ldots ,0)} = F_{(0,0,\ldots ,0)} = {{\mathrm{Int}}}(Q)\).
For \(\lambda \in \{-\,2,-\,1,1,2\}^n\) define
Note that \(R_\lambda \) is isometric to \([0,1]^n\) and if \(\lambda \notin \{-\;2,2\}^n\), then \(R_{\lambda }\) is one of the neighbouring cubes of Q with side length equal to half the side length of Q.
Set
where \(\mathbb {1}_{C_{\kappa }}\) is the characteristic function of \(C_{\kappa }\).
Remark 5.2
Observe that f is simply the nearest point projection from \({\mathbf {R}}^n\) onto Q. Since Q is convex it has infinite reach and [18, 4.8(4)(8)] shows that f is Lipschitz continuous. However, we shall need the above decomposition of f to be able to effectively smoothen the singularities, see 5.3.
In the next lemma we construct a smooth mapping from \({\mathbf {R}}^n\) onto \([-\,1,1]^n\). This is achieved by post-composing the nearest point projection f with a smooth function which has zero derivative exactly in the directions in which the derivative of f is undefined.
Lemma 5.3
Let \(e_1\), ..., \(e_n\), f, \(C_{\kappa }\), \(F_{\kappa }\), Q be as in 5.1. Assume \(s : {\mathbf {R}}\rightarrow {\mathbf {R}}\) is a function of class \({\mathscr {C}}^{\infty }\) such that \(\mathrm {D}^i s(-1) = 0\) and \(\mathrm {D}^i s(1) = 0\) for \(i \in {\mathscr {P}}\). Define \(h(x) = \sum _{j=1}^n s(x \bullet e_j) e_j\) for \(x \in {\mathbf {R}}^n\). Then
-
(a)
\(g = h \circ f\) is of class \({\mathscr {C}}^\infty \) with \(\mathrm {D}^i g = \mathrm {D}^i h \circ f\) for \(i \in {\mathscr {P}}\).
-
(b)
If s is monotone increasing and \(s(t) = t\) for \(t \in \{-\,2,-\,1,0,1,2\}\), then for each \(\kappa \in \{-\,1,0,1\}^n\) and \(\lambda \in \{-\,2,-\,1,1,2\}^n\) if \(C_{\kappa }\), \(F_{\kappa }\), \(T_{\kappa }\), \(c_{\kappa }\), \(R_{\lambda }\) are as in 5.1, then
$$\begin{aligned}&g[C_{\kappa } ]= F_{\kappa }, \quad g|_{F_{\kappa }} : F_{\kappa } \rightarrow F_{\kappa } \quad \text {is a homeomorphism}, \\&\quad g[T_\kappa ]\subseteq T_\kappa , \quad g[R_\lambda ]\subseteq R_\lambda , \quad g[c_{\kappa } + T_{\kappa } ]\subseteq c_{\kappa } + T_{\kappa }. \end{aligned}$$ -
(c)
Let \(\varepsilon \in (0,1)\). Assume s satisfies \(0 \le s'(t) \le 1 + \varepsilon \) and \(|s(t) - t| \le \varepsilon \) for \(t \in {\mathbf {R}}\) and \(s'(t) > 0\) for \(t \in {\mathbf {R}}{{\mathrm{\sim }}}\{-1,1\}\). Then
$$\begin{aligned}&|g(x) - x| \le (1 + \sqrt{n}) \varepsilon \quad \hbox { for}\ x \in Q + {\mathbf {B}}(0,\varepsilon ), \\&\quad {{\mathrm{Lip}}}(g) = {{\mathrm{Lip}}}(h|_{Q}) \le 1 + \varepsilon , \quad {{\mathrm{Lip}}}(g - \mathrm {id}_{{\mathbf {R}}^n}) \le 1, \\&\quad {{\mathrm{Lip}}}(g|_{K} - \mathrm {id}_{K}) < 1 \quad \hbox { for each compact set}\ K \subseteq {{\mathrm{Int}}}Q. \end{aligned}$$
Proof
Since \(s'(1) = 0 = s'(-1)\) we have
First, assume \(x \in C_{\lambda }\) and \(\dim (T_{\lambda }) > 0\). Define \(p = (T_{\lambda })_\natural \) and \(q = (T_{\lambda }^{\perp })_\natural \). Observe that for \(j = 1,2,\ldots ,n\)
Since \(\dim (T_{\lambda }) > 0\) there exists \(j \in \{1,2,\ldots ,n\}\) such that \(\lambda _j = 0\). Let \(u \in {\mathbf {R}}^n\) be such that
Let \(\kappa \in \{-\,1,0,1\}^n\) be defined by
Then \(x + tqu \in C_{\kappa }\) for all \(t \in (0,1)\) because \(C_{\kappa }\) is convex; hence, \(x \in {\mathrm {Clos}}(C_{\kappa })\). Moreover, using (22), (23), (24), we see that for \(j = 1,2,\ldots ,n\)
Therefore, \(T_{\lambda } \subseteq T_{\kappa }\). Recalling \(x \in {\mathrm {Clos}}(C_{\kappa }) \cap C_{\lambda }\), we see that if \(j \in \{1,2,\ldots ,n\}\) and \(\lambda _j \ne \kappa _j\), then \((x \bullet e_j) = \lambda _j\). Thus,
Using (25) we derive
Moreover, since \(pu \in T_{\kappa }\) and \(x+qu \in C_{\kappa }\) and, by (23), \(x+qu+pu \in C_{\kappa }\) we obtain
Thus,
and, recalling (21),
Since \(|\xi | \le {{\mathrm{Lip}}}(f)|u|\) and h is of class \({\mathscr {C}}^1\) we obtain
This shows that \(\mathrm {D}g(x) = \mathrm {D}h(f(x))\) in case \(\dim (T_{\lambda }) > 0\).
Now we shall deal with the case when \(x \in C_{\lambda }\) and \(\dim (T_{\lambda }) = 0\). This means that \(f(x) \in F_{\lambda }\) is one of the vertexes of Q. Since h is of class \({\mathscr {C}}^1\) and \({{\mathrm{Lip}}}(f) < \infty \) (see 5.2) and \(\mathrm {D}h(f(x)) = 0\) by (21), we get
Hence, in this case we also get \(\mathrm {D}g(x) = \mathrm {D}h(f(x))\).
Now we know that \(\mathrm {D}g(x) = \mathrm {D}h(f(x))\) for all \(x \in {\mathbf {R}}^n\) and since f is continuous we see that g is of class \({\mathscr {C}}^1\). Repeating the whole argument with \(g = h \circ f\) replaced by \(\mathrm {D}g = \mathrm {D}h \circ f\) and proceeding by induction we see that g is of class \({\mathscr {C}}^\infty \) so (a) is proven.
Item (b) readily follows from the definition of g.
Consider now \(\varepsilon \) and s as in (c). For \(x \in Q + {\mathbf {B}}(0,\varepsilon )\) we have
For \(y \in {\mathbf {R}}^n\) and \(u \in {\mathbf {R}}^n\) with \(|u|=1\)
For any \(y \in Q\) and \(u \in {\mathbf {R}}^n\), recalling \(-1 \le s'(t) - 1 \le \varepsilon < 1\) for \(t \in {\mathbf {R}}\), we have
For \(K \subseteq {{\mathrm{Int}}}Q\) compact and \(y \in K\) we have \(-1< s'(y \bullet e_i) - 1< \varepsilon < 1\) so \(|\mathrm {D}h(y)u - u|^2 < 1\) whenever \(u \in {\mathbf {R}}^n\) satisfies \(|u|=1\). Consequently, \({{\mathrm{Lip}}}(g|_K - \mathrm {id}_{K}) < 1\). \(\square \)
Next, using 5.3, we construct another function which maps some neighbourhood of \(Q=[-1,1]^n\) onto Q and is the identity a bit further away from Q. To this end we put Q inside a convex open set V with smooth boundary and use the distance from V, which is smooth away from the boundary of V, to interpolate between the mapping constructed in 5.3 and the identity.
Corollary 5.4
Let \(n \in {\mathscr {P}}\). For each \(\varepsilon \in (0,1)\) there exists a map \(l : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) of class \({\mathscr {C}}^\infty \) such that if \(\Gamma = 16 \sqrt{n}\), then
-
(a)
\(l(x) = x\) for \(x \in {\mathbf {R}}^n\) satisfying \({{\mathrm{dist}}}(x,Q) > \varepsilon \).
-
(b)
For each \(\kappa \in \{-1,0,1\}^n\) and \(\lambda \in \{-2,-1,1,2\}^n\) if \(C_{\kappa }\), \(F_{\kappa }\), \(T_{\kappa }\), \(c_{\kappa }\), \(R_{\lambda }\) are as in 5.1, then
$$\begin{aligned}&l[T_{\kappa } ]\subseteq T_{\kappa }, \quad l[F_{\kappa } ]\subseteq F_{\kappa }, \quad l[C_{\kappa } ]\subseteq C_{\kappa }, \quad l[c_{\kappa } + T_{\kappa } ]\subseteq c_{\kappa } + T_{\kappa }, \\&\quad l[R_{\lambda } ]\subseteq R_{\lambda }, \quad l[\{ x \in C_{\kappa } : {{\mathrm{dist}}}(x, Q) \le \varepsilon /\Gamma \} ]\subseteq F_{\kappa }. \end{aligned}$$ -
(c)
\(l|_{{{\mathrm{Int}}}Q} : {{\mathrm{Int}}}Q \rightarrow {{\mathrm{Int}}}Q\) is a diffeomorphism such that for each compact set \(K \subseteq {{\mathrm{Int}}}Q\) we have \({{\mathrm{Lip}}}(l|_K - \mathrm {id}_{K}) < 1\).
-
(d)
\({{\mathrm{Lip}}}(l|_Q - \mathrm {id}_{Q}) \le 1\) and \({{\mathrm{Lip}}}(l|_Q) \le 1 + \varepsilon \).
-
(e)
\({{\mathrm{Lip}}}(l) < \Gamma \).
-
(f)
\(|l(x) - x| \le \varepsilon \) for \(x \in {\mathbf {R}}^n\).
-
(g)
\({{\mathrm{dist}}}(l(x), Q) \le {{\mathrm{dist}}}(x,Q)\) for \(x \in {\mathbf {R}}^n\).
Proof
Let \(n \in {\mathscr {P}}\) and \(\varepsilon \in (0,1)\). Set \(\iota = \varepsilon /(2(1 + \sqrt{n}))\). Let \(\alpha : {\mathbf {R}}\rightarrow {\mathbf {R}}\) be map of class \({\mathscr {C}}^\infty \) such that
Let \(s : {\mathbf {R}}\rightarrow {\mathbf {R}}\) be a homeomorphism of class \({\mathscr {C}}^\infty \) such that
Choose an open convex set \(V \subseteq {\mathbf {R}}^n\) such that \(Q + {\mathbf {B}}(0,\iota /4) \subseteq V \subseteq Q + {\mathbf {B}}(0,\iota /2)\) and \({{\mathrm{Bdry}}}V\) is a submanifold of \({\mathbf {R}}^n\) of class \({\mathscr {C}}^{\infty }\). Define g as in 5.3 using s and set
Since V is convex and \({{\mathrm{Bdry}}}V\) is of class \({\mathscr {C}}^\infty \), we see that l is of class \({\mathscr {C}}^\infty \). Clearly \(l(x) = x\) whenever \({{\mathrm{dist}}}(x,Q) \ge \varepsilon \ge \iota \) which establishes (a).
Proof of (b). Since \(T_\kappa \), \(F_\kappa \), \(C_\kappa \), \(c_\kappa + T_\kappa \), \(R_\lambda \) are convex the inclusions \(l[T_{\kappa } ]\subseteq T_{\kappa }\), \(l[F_{\kappa } ]\subseteq F_{\kappa }\), \(l[C_{\kappa } ]\subseteq C_{\kappa }\), \(l[c_{\kappa } + T_{\kappa } ]\subseteq c_{\kappa } + T_{\kappa }\), \(l[R_{\lambda } ]\subseteq R_{\lambda }\) readily follow from 5.3(b). For \(x \in V\) we have \(l(x) = g(x)\) so, noting \(\iota /4 \ge \varepsilon /(16 \sqrt{n})\), we see that \(l(x) = g(x) \in F_{\kappa }\) whenever \(x \in C_{\kappa }\) and \({{\mathrm{dist}}}(x, Q) \le \varepsilon /\Gamma \).
Employing [18, 4.8(3)], we see that \({{\mathrm{Lip}}}(\delta ) = 1\); hence, we obtain for \(x \in Q + {\mathbf {B}}(0,\varepsilon )\)
Items (c) and (d) follow immediately from 5.3(c) noting that \(l(x) = g(x)\) for \(x \in Q\). Recalling 5.3(c) we have
so (e) holds. For \(x \in {\mathbf {R}}^n\) with \({{\mathrm{dist}}}(x,Q) \le \varepsilon \) we have
which proves (f). To verify (g) note that \(l(x) \in {{\mathrm{conv}}}\{x,g(x)\}\) and \(g(x) \in Q\) for \(x \in {\mathbf {R}}^n\). \(\square \)
6 Central projections
Here we study analytic properties of the central projection from the origin onto the boundary of a bounded convex set V containing 0. In 6.4 we derive formulas and estimates for the derivative of such projection in terms of the position of the origin with respect to the boundary \({{\mathrm{Bdry}}}V\) and the shape of \({{\mathrm{Bdry}}}V\). Then in 6.5 we interpolate between a central projection and the identity to get a map which acts as the central projection inside V and is the identity outside given neighbourhood of V.
We start by deriving a formula for the derivative of the central projection onto the boundary of a half-space.
Remark 6.1
Let \(\nu , y \in {\mathbf {R}}^n\) be such that \(|\nu | = 1\) and \(\nu \bullet y > 0\). Define
Then s and \(\pi \) are maps of class \({\mathscr {C}}^{\infty }\) and \(\pi \) is the central projection from the origin onto the plane \(y + T\). A straightforward computation shows also that for \(z \in U\), \(u \in {\mathbf {R}}^n\)
Definition 6.2
Let \(V \subseteq {\mathbf {R}}^n\) be an open bounded convex set such that \(0 \in V\). We say that a pair of maps \(p : {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\} \rightarrow {\mathbf {R}}^n\) and \(t : {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\} \rightarrow (0,\infty )\) defines the central projection onto \({{\mathrm{Bdry}}}V\) if
In the next lemma we prove that the derivative at some point x of the central projection onto the boundary of a convex set V depends only on the affine tangent plane of \({{\mathrm{Bdry}}}V\) at x (assuming it exists) and, actually, coincides with the derivative of the central projection onto that tangent plane.
Lemma 6.3
Let \(V \subseteq {\mathbf {R}}^n\) be an open bounded convex set with \(0 \in V\). Assume \(y \in {{\mathrm{Bdry}}}V\), and \({{\mathrm{Tan}}}({{\mathrm{Bdry}}}V, y) \in {\mathbf {G}}(n,n-1)\), and \(\nu \in {{\mathrm{Tan}}}({{\mathrm{Bdry}}}V, y)^{\perp }\) is the outward pointing unit normal to \({{\mathrm{Bdry}}}V\) at y; in particular \(|\nu | = 1\) and \(\nu \bullet y > 0\). Suppose U, s, \(\pi \) are defined as in 6.1 and p, t define the central projection onto \({{\mathrm{Bdry}}}V\).
If \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\) satisfies \(p(x) = y\), then p is differentiable at x and
Proof
Fix \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\) such that \(p(x) = y\). Set
Let \(\delta \in {\mathbf {R}}\) satisfy \(0 < \delta \le 2^{-10} (\eta \bullet \nu )^{1/2}\). For \(h \in {\mathbf {R}}^n\) with \(|h| \le \delta |x|\) define
and note that
For \(h \in {\mathbf {R}}^n\) with \(|h| \le \delta |x|\) we have
hence, we can define
Next, set
Since \(T = {{\mathrm{Tan}}}({{\mathrm{Bdry}}}V,y) \in {\mathbf {G}}(n,n-1)\) we know that \(\beta (r) \rightarrow 0\) as \(r \downarrow 0\) and there exists \(0< r_0 < 1\) such that \(\beta (r) \le \frac{1}{2} (1 - \theta )\) for \(0< r < r_0\).
Observe that p is continuous at x. If it were not, there would exist a sequence \(h_i \in {\mathbf {R}}^n\) such that \(|h_i| \rightarrow 0\) as \(i \rightarrow \infty \) but \(y_i = p(x+h_i)\) would not converge to \(y = p(x)\). Then \(y_i\) would be in the cone \(\{ tw : t > 0 , |x-w| \le |h_i|\}\) so, since V is bounded, one could choose a subsequence of \(y_i\) which would converge to some point \(y_0 \in S^{\perp }\) and \(y_0 \ne y\). Since V is convex, this would imply that \(\{ ty + (1-t)y_0 : 0 \le t \le 1 \} \subseteq {{\mathrm{Bdry}}}V\). This, in turn, would mean that \(\eta \in T = {{\mathrm{Tan}}}({{\mathrm{Bdry}}}V, y)\) which is impossible because \(|{T^{\perp }}_\natural \eta | = \eta \bullet \nu > 0\).
Knowing that p is continuous we can find \(\rho _0 > 0\) such that \(|p(x+h) - p(x)| \le r_0\) whenever \(|h| \le \rho _0\). Fix \(h \in {\mathbf {R}}^n\) with \(|h| \le \min \{ \delta |x| , \rho _0 \}\) and let \(b = p(x+h)\). Set
We shall show that \(|b-y| = |p(x+h) - p(x)| \le \Gamma |h|\).
Let \(a,z \in {\mathbf {R}}^n\) be such that
Since \((Z^\perp _h)_\natural (b-a) = 0\) and \({S^\perp }_\natural (z-a) = 0\) and \({T^{\perp }}_\natural (b-z) = 0\) we obtain
Thus
Directly from the definition of a it follows that
Recall that \(|h| \le \rho _0\) so \(|b-y| \le r_0 < 1\) so \(\beta (|b-y|) \le \frac{1}{2} (1-\theta )\) and we can write
In consequence
Let \(h \in {\mathbf {R}}^n\) be such that \(|h| \le \min \{ \delta |x|, \rho _0\}\). Now we are ready to estimate \(|p(x+h) - \pi (x+h)|\). Set \(u = p(x+h) - \pi (x+h)\) and observe that
It is clear from the definitions that \(p(x) = \pi (x) = y\). In consequence
which shows that p is differentiable at x and \(\mathrm {D}p(x) = \mathrm {D}\pi (x)\). \(\square \)
Corollary 6.4
Suppose \(V \subseteq {\mathbf {R}}^n\) is an open bounded convex set, and \(0 \in V\), and \({{\mathrm{Bdry}}}V\) is an \((n-1)\) dimensional submanifold of \({\mathbf {R}}^n\) of class \({\mathscr {C}}^\infty \), and p, t define the central projection onto \({{\mathrm{Bdry}}}V\), and \(\nu (y)\) is the outward pointing unit normal to \({{\mathrm{Bdry}}}V\) at y for \(y \in {{\mathrm{Bdry}}}V\). Then p and t are of class \({\mathscr {C}}^\infty \) and
for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\) and \(u \in {\mathbf {R}}^n\).
Proof
Since \({{\mathrm{Bdry}}}V\) is of class \({\mathscr {C}}^1\) we can apply 6.3 and 6.1 at any single point \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\) to see that
for \(u \in {\mathbf {R}}^n\). Noting \(t(x) = p(x) \bullet x / (x \bullet x)\) one derives the formula for \(\mathrm {D}t(x)\). This shows that p and t are of class \({\mathscr {C}}^1\). Since \(\nu \) is of class \({\mathscr {C}}^{\infty }\), proceeding by induction, we see that p and t are of class \({\mathscr {C}}^\infty \). \(\square \)
Next, we construct a map which interpolates between the central projection onto \({{\mathrm{Bdry}}}V\) and identity outside some neighbourhood of V.
Corollary 6.5
Let \(\varepsilon \in (0,1)\) and \(V \subseteq {\mathbf {R}}^n\) be open convex with \(0 \in V\) and p, t define the central projection onto \({{\mathrm{Bdry}}}V\). Assume \({{\mathrm{Bdry}}}V\) is an \(n-1\) dimensional submanifold of \({\mathbf {R}}^n\) of class \({\mathscr {C}}^\infty \). Then there exists a map \(q : {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\} \rightarrow {\mathbf {R}}^n\) of class \({\mathscr {C}}^\infty \) such that
-
(a)
\(q(x) = x\) for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}V\).
-
(b)
\(q(x) = p(x)\) for \(x \in V {{\mathrm{\sim }}}\{0\}\) with \({{\mathrm{dist}}}(x, {\mathbf {R}}^n {{\mathrm{\sim }}}V) \ge \varepsilon \).
-
(c)
For each \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\) there exists \(t \in [1,\infty )\) such that \(q(x) = tx\).
-
(d)
\(q(x) \in {{\mathrm{conv}}}\{ x, p(x)\}\) for each \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\).
-
(e)
\(|q(x) - x| \le |p(x) - x|\) whenever \(x \in V {{\mathrm{\sim }}}\{0\}\).
-
(f)
\(\Vert \mathrm {D}q(x)\Vert \le 5 |p(x)| |x|^{-1} \Delta \) for \(x \in V {{\mathrm{\sim }}}\{0\}\), where \(\Delta = \inf \bigl \{ \nu (y) \bullet \frac{y}{|y|} : y \in {{\mathrm{Bdry}}}V \bigr \}^{-1}\).
Proof
Set
Then for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\)
Choose \(\alpha : {\mathbf {R}}\rightarrow {\mathbf {R}}\) of class \({\mathscr {C}}^\infty \) such that
Set \(q(x) = \alpha (t(x)) x\).
Clearly \(q : {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\} \rightarrow {\mathbf {R}}^n\) is of class \({\mathscr {C}}^\infty \) and \(q(x) \in {{\mathrm{conv}}}\{x, p(x)\}\) for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\) and \(q(x) = x\) for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}V\). Moreover, \(q(x) = p(x)\) for \(x \in V {{\mathrm{\sim }}}\{0\}\) satisfying \({{\mathrm{dist}}}(x, {\mathbf {R}}^n {{\mathrm{\sim }}}V) \ge \varepsilon \), which follows by (27). For \(x \in V {{\mathrm{\sim }}}\{0\}\) with \(0 < {{\mathrm{dist}}}(x,{\mathbf {R}}^n {{\mathrm{\sim }}}V) \le \iota \) we have, using (26),
For \(x \in V {{\mathrm{\sim }}}\{0\}\), using the formulas from 6.4 and the identity \(t(x) = |p(x)| |x|^{-1}\), we get
7 Smooth deformation theorem
Here we prove a version of Federer–Fleming projection theorem suited for our purposes. The proof follows the scheme of [19, 4.2.6–9]. A similar result was also proven in [14, Theorem 3.1]. However, we need the deformation to be smooth, we need to work in Whitney cubes rather than in a grid of cubes of the same size, and we also need estimates on the measure of the whole deformation.
To deal with families of dyadic cubes it will be convenient to introduce some more notation. We shall follow [5, 1.1–1.9].
Definition 7.1
Let \(k \in \{0,1,\ldots ,n\}\) and \(Q = [0,1]^k \subseteq {\mathbf {R}}^k\). We say that \(R \subseteq {\mathbf {R}}^n\) is a cube if there exist \(p \in {\mathbf {O}}^*({n},{k})\) and \(o \in {\mathbf {R}}^n\) and \(l \in (0,\infty )\) such that \(R = \varvec{\tau }_{o} \circ p^* \circ \varvec{\mu }_{l} [Q ]\). We call \({\mathbf {o}}(R) = o\) the corner of R and \({\mathbf {l}}(R) = l\) the side-length of R. We also set
-
\(\dim (R) = k\)–the dimension of R,
-
\({\mathbf {c}}(R) = {\mathbf {o}}(R) + \frac{1}{2} {\mathbf {l}}(R) (1,1,\ldots ,1)\)–the centre of R,
-
\({\text {Bdry}}_{\mathrm {c}}(R) = \varvec{\tau }_{{\mathbf {o}}(R)} \circ p^* \circ \varvec{\mu }_{{\mathbf {l}}(R)} [{{\mathrm{Bdry}}}Q ]\) – the boundary of R,
-
\({\text {Int}}_{\mathrm {c}}(R) = R {{\mathrm{\sim }}}{\text {Bdry}}_{\mathrm {c}}(R)\)–the interior of R.
Definition 7.2
Let \(k \in \{0,1,\ldots ,n\}\), and \(N \in {\mathbf {Z}}\), and \(Q = [0,1]^k \subseteq {\mathbf {R}}^k\), and \(e_1\), ..., \(e_n\) be the standard basis of \({\mathbf {R}}^n\), and \(f_1\), ..., \(f_k\) be the standard basis of \({\mathbf {R}}^k\).
We define \({\mathbf {K}}_k(N)\) to be the set of all cubes \(R \subseteq {\mathbf {R}}^n\) of the form \(R = \varvec{\tau }_{v} \circ p^* \circ \varvec{\mu }_{2^{-N}} [Q ]\), where \(v \in \varvec{\mu }_{2^{-N}}[{\mathbf {Z}}^n ]\) and \(p \in {\mathbf {O}}^*({n},{k})\) is such that \(p^*(f_i) \in \{ e_1,\ldots ,e_n\}\) for \(i = 1,2,\ldots ,k\).
We also set
Definition 7.3
Let \(k \in \{0,1,\ldots ,n\}\), \(N \in {\mathbf {Z}}\), and \(K \in {\mathbf {K}}_k(N)\). We say that \(L \in {\mathbf {K}}_*\) is a face of K if and only if \(L \subseteq K\) and \(L \in {\mathbf {K}}_j(N)\) for some \(j \in \{0,1,\ldots ,k\}\).
Definition 7.4
(cf. [5, 1.5]) A family of top-dimensional cubes \({\mathcal {F}} \subseteq {\mathbf {K}}\) is said to be admissible if
-
(a)
\(K,L \in {\mathcal {F}}\) and \(K \ne L\) implies \({\text {Int}}_{\mathrm {c}}(K) \cap {\text {Int}}_{\mathrm {c}}(L) = \varnothing \),
-
(b)
\(K,L \in {\mathcal {F}}\) and \(K \cap L \ne \varnothing \) implies \(\frac{1}{2} \le {\mathbf {l}}(L) / {\mathbf {l}}(K) \le 2\),
-
(c)
\(K \in {\mathcal {F}}\) implies \({\text {Bdry}}_{\mathrm {c}}(K) \subseteq \bigcup \{ L \in {\mathcal {F}} : L \ne K \}\).
Definition 7.5
(cf. [5, 1.6]) Let \(U \subseteq {\mathbf {R}}^n\) be an open set and \(e_1\), ..., \(e_n\) be the standard basis of \({\mathbf {R}}^n\). We define the Whitney family \(\mathbf {WF}(U)\) corresponding to U to consist of those cubes \(K \in {\mathbf {K}}\) for which
-
\({{\mathrm{dist}}}_{\infty }(K,{\mathbf {R}}^n {{\mathrm{\sim }}}U) > 2 {\mathbf {l}}(K)\),
-
if \(K \subseteq L \in {\mathbf {K}}\) and \({\mathbf {l}}(L) = 2 {\mathbf {l}}(K)\), then \({{\mathrm{dist}}}_{\infty }(L,{\mathbf {R}}^n {{\mathrm{\sim }}}U) \le 4 {\mathbf {l}}(K)\).
where \({{\mathrm{dist}}}_{\infty }(x,y) = \max \{ | (x-y) \bullet e_i| : i = 1,2,\ldots ,n \}\) for \(x,y \in {\mathbf {R}}^n\) and \({{\mathrm{dist}}}_{\infty }(A,B) = \inf \{ {{\mathrm{dist}}}_{\infty }(x,y) : x \in A,\, y \in B\}\) for \(A,B \subseteq {\mathbf {R}}^n\).
Remark 7.6
If \(U \subseteq {\mathbf {R}}^n\) is open, then the Whitney family \(\mathbf {WF}(U)\) is admissible.
Definition 7.7
(cf. [5, 1.8]) Let \({\mathcal {F}} \subseteq {\mathbf {K}}\) be admissible. We define the cubical complex \(\mathbf {CX}({\mathcal {F}})\) of \({\mathcal {F}}\) to consist of all those cubes \(K \in {\mathbf {K}}_*\) for which
-
K is a face of some cube from \({\mathcal {F}}\),
-
if \(\dim (K) > 0\), then \({\mathbf {l}}(K) \le {\mathbf {l}}(L)\) whenever L is a face of some cube in \({\mathcal {F}}\) with \(\dim (K) = \dim (L)\) and \({\text {Int}}_{\mathrm {c}}(K) \cap {\text {Int}}_{\mathrm {c}}(L) \ne \varnothing \).
Remark 7.8
The second item of 7.7 means that whenever two top dimensional cubes P and Q from \({\mathcal {F}}\) touch and P is smaller than Q and F is a lower dimensional face of Q such that \(Q \cap P \subseteq F\), then the cubical complex \(\mathbf {CX}({\mathcal {F}})\) does not contain F but rather cubes coming from subdivision of F. This is a key property allowing to construct deformations onto skeletons of \(\mathbf {CX}(\mathcal F)\).
Now we are ready to construct a map which is the main building block for the deformation Theorem 7.13. Given \(a \in (-\,1,1)^n\) we need to construct a \({\mathscr {C}}^\infty \) smooth function \({\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) which maps the pointed cube \(Q {{\mathrm{\sim }}}\{a\} = [-1,1]^n {{\mathrm{\sim }}}\{a\}\) onto \({{\mathrm{Bdry}}}Q\), preserves all the lower dimensional skeletons of Q, preserves the neighbouring dyadic cubes of side length 1, is very close to the identity on \({{\mathrm{Bdry}}}Q\), and is the identity outside a small neighbourhood of Q. Moreover, if \({{\mathrm{dist}}}(a,{{\mathrm{Bdry}}}Q) \ge 1/2\), then we need to control the derivative at \(x \in Q {{\mathrm{\sim }}}\{a\}\) of this function by a quantity of magnitude \(|x-a|^{-1}\). To achieve all this we proceed as follows. First we apply a diffeomorphism of \({\mathbf {R}}^n\) which preserves Q and moves a onto the origin–this step is necessary to preserve the neighbouring cubes. Then, we use the “central projection” constructed in 6.5 to map \(Q {{\mathrm{\sim }}}\{a\}\) onto the boundary of some convex set V with smooth boundary such that \(Q \subseteq V\). Finally, we employ the “smooth retraction” produced in 5.4 to map \({{\mathrm{Bdry}}}V\) onto \({{\mathrm{Bdry}}}Q\).
Lemma 7.9
Let \(Q = [-1,1]^n\) and \(\varepsilon \in \bigl (0,\frac{1}{4}\bigr )\). There exists \(\Gamma = \Gamma (n) \in (0,\infty )\) such that for each \(a \in {{\mathrm{Int}}}(Q)\) there exists \(\varphi _{a,\varepsilon } : {\mathbf {R}}^n {{\mathrm{\sim }}}\{a\} \rightarrow {\mathbf {R}}^n\) of class \({\mathscr {C}}^\infty \) such that
-
(a)
\(\varphi _{a,\varepsilon }(x) = x\) for \(x \in {\mathbf {R}}^n\) with \({{\mathrm{dist}}}(x,Q) \ge \varepsilon \).
-
(b)
\(|\varphi _{a,\varepsilon }(x) - x| \le \varepsilon \) for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}Q\).
-
(c)
\(\varphi _{a,\varepsilon } [Q {{\mathrm{\sim }}}\{a\} ]= {{\mathrm{Bdry}}}Q\).
-
(d)
If \(\kappa \in \{-1,0,1\}^n {{\mathrm{\sim }}}\{ (0,0,\ldots ,0) \}\) and \(\lambda \in \{-2,-1,1,2\}^n {{\mathrm{\sim }}}\{-2,2\}^n \) and \(C_{\kappa }\), \(F_{\kappa }\), \(T_{\kappa }\), \(c_{\kappa }\), \(R_\lambda \) are defined as in 5.1, then
$$\begin{aligned}&\varphi _{a,\varepsilon }[F_{\kappa } ]\subseteq {\mathrm {Clos}}F_{\kappa }, \quad \varphi _{a,\varepsilon }[C_{\kappa } ]\subseteq {\mathrm {Clos}}C_{\kappa }, \quad \varphi _{a,\varepsilon }[c_{\kappa } + T_{\kappa } ]\subseteq c_{\kappa } + T_{\kappa }, \\&\quad \varphi _{a,\varepsilon }[R_{\lambda } ]\subseteq R_{\lambda }, \quad \varphi _{a,\varepsilon }[T_{\kappa } {{\mathrm{\sim }}}Q ]\subseteq T_{\kappa }, \quad a \in T_{\kappa } \quad \text {implies} \quad \varphi _{a,\varepsilon }[T_{\kappa } {{\mathrm{\sim }}}\{a\} ]\subseteq T_{\kappa }. \end{aligned}$$ -
(e)
For \(x \in {{\mathrm{Int}}}Q {{\mathrm{\sim }}}\{a\}\) we have
$$\begin{aligned} \Vert \mathrm {D}\varphi _{a,\varepsilon }(x)\Vert \le \frac{\Gamma }{2 |x-a| {{\mathrm{dist}}}(a, {\mathbf {R}}^n {{\mathrm{\sim }}}Q)}. \end{aligned}$$ -
(f)
For \(x \in {\mathbf {R}}^n\) with \({{\mathrm{dist}}}(x, {{\mathrm{Bdry}}}Q) \le \min \left\{ \frac{1}{2} {{\mathrm{dist}}}(a, {\mathbf {R}}^n {{\mathrm{\sim }}}Q), \frac{1}{4} \right\} \) there holds
$$\begin{aligned} \Vert \mathrm {D}\varphi _{a,\varepsilon }(x)\Vert \le \Gamma . \end{aligned}$$ -
(g)
For \(x \in {\mathbf {R}}^n\) with \({{\mathrm{dist}}}(x, {{\mathrm{Bdry}}}Q) \le \min \left\{ \frac{1}{2} {{\mathrm{dist}}}(a, {\mathbf {R}}^n {{\mathrm{\sim }}}Q), \frac{1}{4} \right\} \) we obtain
$$\begin{aligned} |\varphi _{a,\varepsilon }(x) - x| \le \Gamma ({{\mathrm{dist}}}(x,{{\mathrm{Bdry}}}Q) + \varepsilon ). \end{aligned}$$ -
(h)
If \(x \in {\mathbf {R}}^n\), \(\delta \in {\mathbf {R}}\), \(0< \delta < \min \left\{ \frac{1}{2} {{\mathrm{dist}}}(a, {\mathbf {R}}^n {{\mathrm{\sim }}}Q), \frac{1}{4} \right\} \), and \({{\mathrm{dist}}}(x, {{\mathrm{Bdry}}}Q) \le \delta \), then
$$\begin{aligned} {{\mathrm{conv}}}\bigl \{ x, \varphi _{a,\varepsilon }(x) \bigr \} \subseteq {{\mathrm{Bdry}}}Q + {\mathbf {B}}(0,\delta ). \end{aligned}$$In particular \({{\mathrm{dist}}}(\varphi _{a,\varepsilon }(x), {{\mathrm{Bdry}}}Q) \le {{\mathrm{dist}}}(x, {{\mathrm{Bdry}}}Q)\) for each \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{a\}\).
-
(i)
Let \(X \subseteq {{\mathrm{Bdry}}}Q\) be compact and convex, \(\kappa \in \{-1,0,1\}^n\), \(T_{\kappa }\) be defined as in 5.1. For \(a \in {{\mathrm{Int}}}Q\) define \(E_a = \varphi _{a,\varepsilon }^{-1}[X ]\). Then
$$\begin{aligned} \lim _{b \rightarrow a} {d_{{\mathscr {H}}}}\bigl ( E_a, E_b \bigr ) = 0 \quad \hbox { for}\ a \in {{\mathrm{Int}}}Q. \end{aligned}$$(28)Moreover, if \(\mu \) is a Radon measure over \({\mathbf {R}}^n\) and \(\dim T_{\kappa } = k\), then for \({\mathscr {H}}^k\) almost all \(a \in \bigl (-\tfrac{1}{2}, \tfrac{1}{2} \bigr )^n \cap T_{\kappa }\)
$$\begin{aligned} \mu \bigl ( {\textstyle \bigcap _{\delta > 0} \bigcup _{b \in {\mathbf {B}}(a,\delta ) \cap T_{\kappa }}} ((E_a {{\mathrm{\sim }}}E_b) \cup (E_b {{\mathrm{\sim }}}E_a)) \cap T_{\kappa } \bigr ) = 0. \end{aligned}$$
Proof
For each \(\delta \in [0,\infty )\) let \(p_\delta \), \(t_\delta \) define the central projection onto \({{\mathrm{Bdry}}}( Q + {\mathbf {U}}(0,\delta ))\) as in 6.2. Observe that
-
\(t_\delta \) is continuous for each \(\delta \in [0,\infty )\),
-
\(t_\delta (x) \le t_\sigma (x)\) whenever \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\) and \(0 \le \delta \le \sigma < \infty \),
-
\(\lim _{\delta \downarrow 0} t_\delta (x) = t_0(x)\) for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\).
Employing the Dini Theorem (cf. [38, 7.13]), we see that \(t_\delta \) converge uniformly to \(t_0\) as \(\delta \downarrow 0\) on compact subsets of \({\mathbf {R}}^n {{\mathrm{\sim }}}\{0\}\). Therefore, there exists \(\delta _0 \in (0,\varepsilon )\) such that
Set \(\iota = \min \{ \delta _0, 2^{-10} \varepsilon / \Gamma _{{5.4}}\}\). Choose an open convex set \(V \subseteq {\mathbf {R}}^n\) with \({\mathscr {C}}^\infty \) smooth boundary and such that \(Q + {\mathbf {B}}(0,\iota /4) \subseteq V \subseteq Q + {\mathbf {B}}(0,\iota )\). Such V is easily constructed, e.g., by taking first the set \({{\tilde{V}}} = Q + {\mathbf {B}}(0,\iota /2)\), representing \({{\mathrm{Bdry}}}{{\tilde{V}}}\) locally as (rotated) graph of some convex function, and then mollifying this function. Employ 5.4 with \(2^{-10} \varepsilon \) in place of \(\varepsilon \) to obtain the map \(l \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n,{\mathbf {R}}^n)\). Apply 6.5 with \(2^{-10}\iota \), V in place of \(\varepsilon \), V to construct the map \(q \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n {{\mathrm{\sim }}}\{0\},{\mathbf {R}}^n)\).
Fix a symmetric non-negative mollifier \(\phi \in {\mathscr {C}}^{\infty }({\mathbf {R}},{\mathbf {R}})\) whose support equals \([-1,1]\) and set \(\phi _{\rho }(s) = \rho ^{-1} \phi (s/\rho )\) for \(s \in {\mathbf {R}}\) and \(\rho > 0\). Let \(e_1,\ldots ,e_n\) be the standard basis of \({\mathbf {R}}^n\). For \(i = 1,2,\ldots ,n\) set \(\rho _i = \min \bigl \{ \frac{1}{2}, 1 - |a \bullet e_i| \bigr \}\) and let \({\bar{f}}_{a,i} : {\mathbf {R}}\rightarrow {\mathbf {R}}\) be the continuous piece-wise affine map satisfying
Next, define \(f_{a,i} = {\bar{f}}_{a,i} * \phi _{\rho _i/8} \in {\mathscr {C}}^{\infty }({\mathbf {R}},{\mathbf {R}})\). We obtain
-
if \(a \bullet e_i = 0\), then \(f_{a,i}(t) = t\) for \(t \in {\mathbf {R}}\);
-
if \(|a \bullet e_i| > 0\), then
$$\begin{aligned}&f_{a,i}(a \bullet e_i) = 0, \quad f_{a,i}(t) = t \quad \hbox {for}\;t \in {\mathbf {R}}\;\hbox {with}\; |t| \ge 1 - \tfrac{1}{2} \rho _i, \\&\quad \frac{1}{2(2-\rho _i)}< f_{a,i}'(t) < \frac{2}{\rho } \quad \text {and} \quad |f_i(t)| \ge \tfrac{1}{2} |t - a \bullet e_i| \quad \hbox { for}\;\ t\;\in {\mathbf {R}}. \end{aligned}$$
Define the map \(f_a : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) by \(f_a(x) = \sum _{i=1}^n f_{a,i}(x \bullet e_i) e_i\). Then
Define \(\varphi _{a,\varepsilon } = l \circ q \circ f_a\). Clearly \(\varphi _{a,\varepsilon } : {\mathbf {R}}^n {{\mathrm{\sim }}}\{a\} \rightarrow {\mathbf {R}}^n\) is of class \({\mathscr {C}}^\infty \).
Proof of (a): If \(x \in {\mathbf {R}}^n\) satisfies \({{\mathrm{dist}}}(x, Q) \ge \varepsilon \), then \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}V\), so \(q(f_a(x)) = q(x) = x\) and \(\varphi _{a,\varepsilon }(x) = l(x) = x\).
Proof of (b): Let \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}Q\). Then \(f_a(x) = x\). If \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}V\), then \(q(x) = x\) and \(|\varphi _{a,\varepsilon }(x) - x| = |l(x) - x| \le 2^{-10} \varepsilon \) by 5.4(f). Let p, t define the central projection onto \({{\mathrm{Bdry}}}V\) as in 6.2. If \(x \in V {{\mathrm{\sim }}}Q\), then by 6.5(e) and (29) we have
Hence, \(|\varphi _{a,\varepsilon }(x) - x| \le |l(q(x)) - q(x)| + |q(x) - x| \le \varepsilon \) by 5.4(f).
Proof of (c): If \(x \in Q {{\mathrm{\sim }}}\{a\}\), then \(f_a(x) \in Q {{\mathrm{\sim }}}\{0\}\) and \({{\mathrm{dist}}}(f_a(x),{\mathbf {R}}^n {{\mathrm{\sim }}}V) \ge \iota /4\), so \(q(f_a(x)) \in {{\mathrm{Bdry}}}V\) by 6.5(b). Consequently \({{\mathrm{dist}}}(q(f_a(x)),Q) \le \iota \le 2^{-10}\varepsilon / \Gamma _{{5.4}}\), which implies \(\varphi _{a,\varepsilon }(x) = l(q(f_a(x))) \in {{\mathrm{Bdry}}}V\) by 5.4(b).
Proof of (d): For \(\kappa \in \{-1,0,1\}^n\) let \(F_{\kappa }\), \(C_{\kappa }\), \(T_{\kappa }\), and \(c_{\kappa }\) be defined as in 5.1. Fix a \(\kappa \in \{-1,0,1\}^n\) with \(\kappa \ne (0,\ldots ,0)\). Let \(x \in F_{\kappa }\) and note that \(f_a(x) = x\). Recall \(\bigcup \{ C_{\lambda } : \lambda \in \{-1,0,1\}^n {{\mathrm{\sim }}}\{(0,\ldots ,0)\} \} = {\mathbf {R}}^n {{\mathrm{\sim }}}{{\mathrm{Int}}}Q\), so there exists \(\lambda \in \{-1,0,1\}^n\) such that \(q(x) \in C_{\lambda }\). Since \(q(x) = tx\) for some \(t > 1\), it follows from the definition of \(C_{\lambda }\) and \(C_{\kappa }\) that \(\lambda _j = \kappa _j\) whenever \(\kappa _j \ne 0\), which implies that \(F_{\lambda } \subseteq {\mathrm {Clos}}F_{\kappa }\). Since \(l[F_{\lambda } ]\subseteq F_{\lambda }\) we get \(\varphi _{a,\varepsilon }(x) = l(q(x)) \in {\mathrm {Clos}}(F_{\kappa })\). Similarly, if \(x \in C_{\kappa }\), then \(f_a(x) = x\) and there exists \(\lambda \in \{-1,0,1\}^n\) such that \(q(x) \in C_{\lambda }\) and \(q(x) = tx\) for some \(t \ge 1\); hence, \(C_{\lambda } \subseteq {\mathrm {Clos}}C_{\kappa }\) and \(\varphi _{a,\varepsilon }(x) = l(q(x)) \in {\mathrm {Clos}}(C_{\kappa })\) because \(l[C_{\lambda } ]\subseteq C_{\lambda }\).
For \(y \in (c_{\kappa } + T_{\kappa }) {{\mathrm{\sim }}}V\) we have \(q(f_a(y)) = q(y) = y\) and then \(\varphi _{a,\varepsilon }(y) = l(y) \in c_{\kappa } + T_{\kappa }\) by 5.4(b). If \(y \in (c_{\kappa } + T_{\kappa }) \cap V\), then \(q(f_a(y)) = q(y) = ty\) for some \(t \ge 1\), by 6.5(d), and, as before, \(q(y) \in C_{\lambda }\) for some \(\lambda \in \{-1,0,1\}^n\) such that \(\lambda _j = \kappa _j\) whenever \(\kappa _j \ne 0\). In this case \({{\mathrm{dist}}}(y,Q) \le \varepsilon /\Gamma _{{5.4}}\) and we can apply 5.4(b) to see that \(l(q(y)) \in F_{\lambda } \subseteq {\mathrm {Clos}}F_{\kappa } \subseteq c_{\kappa } + T_{\kappa }\).
Assume now \(\lambda \in \{-\,2,-\,1,1,2\}^n {{\mathrm{\sim }}}\{-\,2,2\}^n\) and \(x \in R_\lambda \). If \(x \notin V\), then \(\varphi _{a,\varepsilon }(x) = l(x) \in R_\lambda \) by 5.4(b). Thus, assume \(x \in R_\lambda \cap V\). Let \(\kappa \in \{-1,0,1\}^n\) be such that \(\kappa _i = \lambda _i\) if \(|\lambda _i| = 1\) and \(\kappa _i = 0\) if \(|\lambda _i| = 2\) for \(i = 1,2,\ldots ,n\). We already know that \(l(q(x)) \in {\mathrm {Clos}}F_\kappa \) so it suffices to show that \((q(x) \bullet e_j) \lambda _j \ge 0\) whenever \(|\lambda _j| = 2\) for some \(j \in \{1,\ldots ,n\}\) but this is clear since \(q(x) = tx\) for some \(t > 0\).
If \(a \in T_{\kappa }\), then \(f_a[T_{\kappa } {{\mathrm{\sim }}}\{a\} ]\subseteq T_{\kappa } {{\mathrm{\sim }}}\{0\}\). For \(x \in T_{\kappa } {{\mathrm{\sim }}}\{0\}\) we have \(q(x) = tx\) for some \(t \in [1,\infty )\), so \(q[T_{\kappa } {{\mathrm{\sim }}}\{0\} ]\subseteq T_{\kappa }\). Finally, \(l[T_{\kappa } {{\mathrm{\sim }}}\{0\} ]\subseteq T_{\kappa }\); hence, \(\varphi _{a,\varepsilon }[T_{\kappa } {{\mathrm{\sim }}}\{0\} ]\subseteq T_{\kappa }\). If \(x \in T_\kappa {{\mathrm{\sim }}}Q\), then \(f_a(x) = x\) so \(\varphi _{a,\varepsilon }[T_{\kappa } {{\mathrm{\sim }}}Q ]\subseteq T_{\kappa }\) as before.
Proof of (e): Set \(\Delta = \inf \bigl \{\nu (y) \bullet \tfrac{y}{|y|} : y \in {{\mathrm{Bdry}}}V \bigr \}^{-1}\). Note that \(\Delta \) can be bounded by a constant depending only on n and, in particular, independently of \(\varepsilon \). Employ 6.5(f) to get for \(z \in V {{\mathrm{\sim }}}\{0\}\)
Hence, combining 5.4(e), (30), and (31), for \(x \in Q {{\mathrm{\sim }}}\{a\}\)
Proof of (f): If \(z \in {\mathbf {R}}^n {{\mathrm{\sim }}}V\), then \(\Vert \mathrm {D}q(z)\Vert = 1\) and \(|z| > 1\). If \(z \in V\) and \({{\mathrm{dist}}}(z,{{\mathrm{Bdry}}}Q) \le \frac{1}{4}\), then \(|z| \ge \frac{3}{4}\), so \(\Vert \mathrm {D}q(z)\Vert \le 15 \Delta \). Altogether, for \(x \in {\mathbf {R}}^n\) satisfying
we have \(f_a(x) = x\) and, by 5.4(e),
Proof of (g): Let \(x \in {\mathbf {R}}^n\) satisfy \({{\mathrm{dist}}}(x, {{\mathrm{Bdry}}}Q) \le \min \left\{ \frac{1}{2} {{\mathrm{dist}}}(a, {\mathbf {R}}^n {{\mathrm{\sim }}}Q), \frac{1}{4} \right\} \). Let \(y \in {\mathbf {R}}^n\) be such that \({{\mathrm{dist}}}(y,Q) = \varepsilon \) and \(|x-y| \le {{\mathrm{dist}}}(x,{{\mathrm{Bdry}}}Q) + \varepsilon \). Then, \(\Vert \mathrm {D}\varphi _{a,\varepsilon }(tx + (1-t)y)\Vert \le 15 \Delta \Gamma _{{5.4}}\) for each \(t \in [0,1]\) and \(\varphi _{a,\varepsilon }(y) = y\), so
Proof of (h): Let \(x \in {\mathbf {R}}^n\), \(\delta \in {\mathbf {R}}\), \(0< \delta < \min \left\{ \frac{1}{2} {{\mathrm{dist}}}(a, {\mathbf {R}}^n {{\mathrm{\sim }}}Q), \frac{1}{4} \right\} \), and \({{\mathrm{dist}}}(x, {{\mathrm{Bdry}}}Q) \le \delta \); then \(f_a(x) = x\).
In case \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}Q\), if \(\kappa \in \{-1,0,1\}^n\) is such that \(x \in C_{\kappa }\), then \({{\mathrm{dist}}}(x, Q) = {{\mathrm{dist}}}(x,F_{\kappa })\) and \(\varphi _{a,\varepsilon }(x) \in {\mathrm {Clos}}C_{\kappa }\). Since \({\mathrm {Clos}}C_{\kappa } \cap ({\mathrm {Clos}}F_{\kappa } + {\mathbf {B}}(0,\delta )) \subseteq {{\mathrm{Bdry}}}Q + {\mathbf {B}}(0,\delta )\) is convex and contains both x and \(\varphi _{a,\varepsilon }(x)\), we see that \({{\mathrm{conv}}}\{x,\varphi _{a,\varepsilon }(x)\} \subseteq {{\mathrm{Bdry}}}Q + {\mathbf {B}}(0,\delta )\).
Assume now \(x \in Q\). Observe that there exists \(\kappa \in \{-1,0,1\}^n\) such that \(\varphi _{a,\varepsilon }(x) \in {\mathrm {Clos}}F_{\kappa }\) and \({{\mathrm{dist}}}(x,{{\mathrm{Bdry}}}Q) = {{\mathrm{dist}}}(x, F_{\kappa })\)–this is because q acts on x as central projection with centre at the origin. As before, we see that x and \(\varphi _{a,\varepsilon }(x)\) both lie in the convex set \({\mathrm {Clos}}C_{\kappa } \cap ({\mathrm {Clos}}F_{\kappa } + {\mathbf {B}}(0,\delta ))\) so \({{\mathrm{conv}}}\{x,\varphi _{a,\varepsilon }(x)\} \subseteq {{\mathrm{Bdry}}}Q + {\mathbf {B}}(0,\delta )\).
Proof of (i): Set \(Y = (l \circ q)^{-1} [X ]\), \(P = (-1/2,1/2)^n\), \(R = (-3/4,3/4)^n\). Observe that \(E_a = \varphi _{a,\varepsilon }^{-1} [X ]= f_a^{-1} [Y ]\) and recall that \(f_a\) is a diffeomorphism for each \(a \in {{\mathrm{Int}}}Q\). It follows from the construction that for any compact set \(K \in {{\mathrm{Int}}}Q\)
For \(a \in {{\mathrm{Int}}}Q\) let \(B_a\) be the topological boundary of \(E_a \cap T_{\kappa }\) relative to \(T_{\kappa }\). Let \(a \in P \cap T_{\kappa }\). It follows from (28) and the construction that
Without loss of generality we may assume \(T_{\kappa } = {{\mathrm{span}}}\{e_1, \ldots , e_k\}\). Recall the construction of the maps l and q to see that Y is a convex conical cap over \(l^{-1}[X ]\) with vertex at the origin. Let B be the topological boundary of \(Y \cap T_{\kappa }\) relative to \(T_{\kappa }\). Define affine lines \(L_{a,i} = \{ a + t e_i : t \in {\mathbf {R}}\}\) for \(a \in {\mathbf {R}}^n\) and \(i \in \{1,2,\ldots ,n\}\). Since B is the boundary of a convex set and has empty interior in \(T_{\kappa }\) we see that there exists \(i \in \{ 1,2,\ldots ,k \}\) such that \(B \cap L_{a,i}\) contains at most two points for each \(a \in T_{\kappa }\) and we may decompose B into two disjoint sets \(B = B_1 \cup B_2\) so that for each \(a \in T_{\kappa }\) if \(a + t_1 e_i \in B_1 \cap L_{i,a}\) and \(a + t_2 e_i \in B_2 \cap L_{i,a}\), then \(t_1 < t_2\). Define \(B_{1,a} = f_{a}^{-1} [B_1 ]\) and \(B_{2,a} = f_{a}^{-1} [B_2 ]\) for \(a \in T_{\kappa }\). Then \(B_a\) equals the disjoint sum of \(B_{1,a}\) and \(B_{2,a}\) for each \(a \in {{\mathrm{Int}}}Q \cap T_{\kappa }\) because \(f_a\) is a diffeomorphism. Clearly, it suffices to show that if \(j \in \{1,2\}\), then for \({\mathscr {H}}^k\) almost all \(a \in P \cap T_{\kappa }\) we have
Fix \(j = \{1,2\}\). Observe that if \(a \in P \cap T_{\kappa }\) and \(t \in (-3/4,3/4)\), then the map \(g_{a,t} : L_{a,i} \cap P \rightarrow {\mathbf {R}}\) given by \(g_{a,t}(b) = f_{b,i}^{-1}(t)\) is strictly increasing. In consequence, if \(b,c \in L_{a,i} \cap P\) and \(b \ne c\), then \(R \cap T_{\kappa } \cap B_{j,b} \cap B_{j,c} = \varnothing \). Since \(\mu \) is Radon, there exists at most countably many \(b \in L_{a,i} \cap P\) for which \(\mu \bigl ( R \cap T_{\kappa } \cap B_{j,b}\bigr ) > 0\). In particular, we obtain for each \(a \in P \cap T_{\kappa }\)
Since 
coincides with the Lebesgue measure \({\mathscr {L}}^k\) on \(T_{\kappa }\) and \({\mathscr {L}}^k\) is the product of k copies of the one dimensional Lebesgue measure (cf. [19, 2.6.5]), we may use the Fubini theorem [19, 2.6.2(3)] to conclude the proof. \(\square \)
The next lemma is a counterpart of [19, 4.2.7]. Given arbitrary Radon measures \(\mu _1, \ldots , \mu _l\), and numbers \(m_1\), ..., \(m_l\), and a k-plane \(T_{\kappa }\) with \(\max \{m_1,\ldots ,m_l\} < k \le n\) we prove that there are enough good points \(a \in [-1/2,1/2]^n \cap T_{\kappa }\) for which the integral \(\int _Q \Vert \mathrm {D}\varphi _{a,\varepsilon }\Vert ^{m_i} \,\mathrm {d}\mu _i\) is controlled by \(\mu _i(Q)\). Later we shall apply this lemma to measures \(\mu \) defined as the restriction of \({\mathscr {H}}^{m}\) to some m dimensional set \(\Sigma \subseteq {\mathbf {R}}^n\) with density.
Lemma 7.10
Suppose
For \(a \in {{\mathrm{Int}}}Q\) let \(\varphi _{a,\varepsilon } : {\mathbf {R}}^n {{\mathrm{\sim }}}\{a\} \rightarrow {\mathbf {R}}^n\) be the map constructed in 7.9 and set
Then \({\mathscr {L}}^k(E) > 0\).
Proof
Employing 7.9(e) we have for \(\varepsilon \in (0,1/4)\), \(m \in (0,k)\), \(x \in Q\), and \(y \in A\) satisfying \(|x-y| = {{\mathrm{dist}}}(x,A)\)
Thus, for \(i \in \{ 1,2,\ldots ,l \}\), using the Fubini theorem [19, 2.6.2], we obtain
Now, we argue by contradiction. If \({\mathscr {L}}^k(E)\) was zero, then we would have
Now, given a cube \(K \in {\mathbf {K}}_*\) (of arbitrary dimension and size) and sets \(\Sigma _1\), ..., \(\Sigma _l\) we combine 7.10 and 7.9 to construct a deformation of \({\mathbf {R}}^n\) which maps \(\Sigma _i \cap K\) into \({\text {Bdry}}_{\mathrm {c}}(K)\) for each \(i = 1,2,\ldots ,l\) and preserves all the super-cubes of K (i.e. those which contain K) as well as all the cubes from \({\mathbf {K}}_*\) which do not touch \({\text {Int}}_{\mathrm {c}}(K)\) and have side length at least \(\frac{1}{2} {\mathbf {l}}(K)\). Of course, we also control the derivative.
Lemma 7.11
Suppose
Then for each \(\varepsilon _0 \in \bigl (0,\frac{1}{4} {\mathbf {l}}(K)\bigr )\) there exist \(\varepsilon \in (0,\varepsilon _0]\), a neighbourhood U of \(\Sigma \) in \({\mathbf {R}}^n\), and a map \(\varphi \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n, {\mathbf {R}}^n)\) such that
-
(a)
\(\varphi \in {\mathfrak {D}}({K + {\mathbf {U}}(0,\varepsilon )})\),
-
(b)
\(\varphi (x) = x\) for \(x \in {\mathbf {R}}^n\) satisfying \({{\mathrm{dist}}}(x,K) \ge \varepsilon \),
-
(c)
\(\varphi [U ]\cap K = \varphi [U \cap K ]\subseteq {\text {Bdry}}_{\mathrm {c}}(K)\),
-
(d)
\(\varphi [{\text {Bdry}}_{\mathrm {c}}(K) + {\mathbf {B}}(0,\varepsilon ) ]\subseteq {\text {Bdry}}_{\mathrm {c}}(K) + {\mathbf {B}}(0,\varepsilon )\),
-
(e)
\(|\varphi (x) - x| \le \varepsilon \) for \(x \in {\mathbf {R}}^n\) satisfying \({T}_\natural (x - {\mathbf {c}}(K)) \notin {T}_\natural [K ]\), where \(T = {{\mathrm{Tan}}}(K,{\mathbf {c}}(K))\),
-
(f)
if \(L \in {\mathbf {K}}_*\) satisfies either \({\mathbf {l}}(L) \ge {\mathbf {l}}(K)\) or \({\mathbf {l}}(L) \ge \frac{1}{2} {\mathbf {l}}(K)\) and \(L \cap {\text {Int}}_{\mathrm {c}}(K) = \varnothing \), then \(\varphi [L ]\subseteq L\).
-
(g)
\(\Vert \mathrm {D}\varphi (x)\Vert \le \Gamma \) for \(x \in {\mathbf {R}}^n\) with \({{\mathrm{dist}}}(x,{\text {Bdry}}_{\mathrm {c}}(K)) \le \varepsilon \),
-
(h)
if \(\max \{m_1, \ldots , m_l\} \le k - 1\), then there exists \(\Gamma = \Gamma (k,l) \in (1,\infty )\) such that
$$\begin{aligned} \int _{K} \Vert \mathrm {D}\varphi \Vert ^{m_i} \,\mathrm {d}\nu _i \le \Gamma \nu _i(K) \quad \hbox { for}\;\ i = 1,2,\ldots ,l. \end{aligned}$$
Proof
Let \(\varepsilon _0 \in (0, {\mathbf {l}}(K)/4)\). If \(\max \{m_1, \ldots , m_l\} \le k - 1\), then set \(\varepsilon = \varepsilon _0\). If \(m_1 = \cdots = m_l = k\), then choose arbitrary \(a_0 \in {\text {Int}}_{\mathrm {c}}(K) {{\mathrm{\sim }}}\Sigma \) and set
Translating \(\Sigma \) and K by \(-{\mathbf {c}}(K)\) we can assume \({\mathbf {c}}(K) = 0\). Set
Note that \(r[K ]= [-1,1]^n \cap T\). For \(a \in K\) let \(\varphi _{r(a), 2\iota /{\mathbf {l}}(K)}\) be the map defined by employing 7.9 with r(a), \(2\iota /{\mathbf {l}}(K)\) in place of a, \(\varepsilon \) and set
To choose an appropriate \(a \in K\), we consider two cases.
-
If \(\max \{m_1, \ldots , m_l\} \le k - 1\), then we proceed as follows. Define \(E \subseteq K\) to be the set of all those \(a \in K\) for which \(r(a) \in [-1/2,1/2]^n\) and
$$\begin{aligned} \int _{[-1,1]^n} \Vert \mathrm {D}\varphi _{r(a),2\iota /{\mathbf {l}}(K)}\Vert ^{m_i} \,\mathrm {d}\mu _i \le l \Gamma _{{7.10}}(k,m_i) \mu _i([-1,1]^n) \quad \hbox { for}\;\ i = 1,\ldots ,l. \end{aligned}$$(32)Apply 7.10 with \(2\iota /{\mathbf {l}}(K)\) in place of \(\varepsilon \) to conclude that \({\mathscr {L}}^k(E) > 0\). Since we have \({\mathscr {H}}^{k}(\Sigma \cap K) = 0\), we may choose \(a \in E {{\mathrm{\sim }}}\Sigma \).
-
If \(m_1 = \cdots = m_l = k\), then we set \(a = a_0\).
Since \(\Sigma \cap K\) is compact we have
Let \(\alpha \in {\mathscr {C}}^{\infty }({\mathbf {R}}, {\mathbf {R}})\) be such that \(\alpha (t) = 1\) for \(t \ge 7/8\), \(\alpha (t) = 0\) for \(t \le 1/4\), and \(0< \alpha '(t) < 2\) for \(t \in (0,1)\). Recall that we assumed \({\mathbf {c}}(K) = 0\); in particular \(K \subseteq T\). Set
Clearly \(\varphi \) is of class \({\mathscr {C}}^\infty \). Since \(K + {\mathbf {U}}(0,\varepsilon )\) is convex, we see that (a) is satisfied. Set
Proof of (b): If \(x \in {\mathbf {R}}^n\) satisfies \({{\mathrm{dist}}}(x,K) \ge \varepsilon \), then either \({{\mathrm{dist}}}(x, T) \ge \varepsilon /\sqrt{2} \ge \iota \) and then \(\varphi (x) = x\), or \({{\mathrm{dist}}}({T}_\natural x, K) \ge \varepsilon / \sqrt{2} \ge \iota \) and then \(\psi _a(x) = x\), by 7.9(a), and \(\varphi (x) = x\).
Proof of (c): If \(x \in U \cap K\), then \(\varphi (x) = \psi _a(x)\). Observe that \(\psi _a(T {{\mathrm{\sim }}}\{a\}) \subseteq T\) by 7.9(d) because \(a \in T\). Combining this with \(\psi _a[R {{\mathrm{\sim }}}\{a\} ]\subseteq {{\mathrm{Bdry}}}R\), which holds due to 7.9(c), we see that \(\varphi [U \cap K ]\subseteq {\text {Bdry}}_{\mathrm {c}}(K)\). Moreover, by 7.9(d) and the definition of \(\varphi \) we have \(\varphi [{\mathbf {R}}^n {{\mathrm{\sim }}}K ]\subseteq {\mathbf {R}}^n {{\mathrm{\sim }}}K\), so \(\varphi [U ]\cap K = \varphi [U \cap K ]\).
Proof of (d): Let \(x \in {\text {Bdry}}_{\mathrm {c}}(K) + {\mathbf {B}}(0,\varepsilon )\). Observe that \(T_\natural ^\perp x = T_\natural ^\perp \varphi (x)\) because \(\psi _a({T}_\natural x) \in T\) due to 7.9(d). Moreover, \({{\mathrm{Bdry}}}R \cap (T + {\mathbf {B}}(0,\varepsilon )) \subseteq {\text {Bdry}}_{\mathrm {c}}(K) + T^{\perp }\), so for any \(z \in {\text {Bdry}}_{\mathrm {c}}(K) + {\mathbf {B}}(0,\varepsilon )\) we have \({{\mathrm{dist}}}(z, {\text {Bdry}}_{\mathrm {c}}(K))^2 = {{\mathrm{dist}}}({T}_\natural z, {{\mathrm{Bdry}}}R)^2 + {{\mathrm{dist}}}(z,T)^2\). Noting \(|T_\natural ^\perp x| \le \varepsilon \) and using 7.9(d, h) we can write
where \(t = \alpha (|T_\natural ^\perp x|/\iota )\). Thus, \(\varphi (x) \in {\text {Bdry}}_{\mathrm {c}}(K) + {\mathbf {B}}(0,\varepsilon )\).
Proof of (e): Let \(x \in {\mathbf {R}}^n\) be such that \({T}_\natural x \notin K\). If \({{\mathrm{dist}}}(x,K) \ge \varepsilon \), then \(\varphi (x) = x\) and there is nothing to prove. Assume \({{\mathrm{dist}}}(x,K) \le \varepsilon \). By 7.9(b) we know \(|\psi ({T}_\natural x) - {T}_\natural x| \le \iota \) so for any \(t \in [0,1]\) we have \(|(t {T}_\natural x + (1-t) \psi ({T}_\natural x)) - {T}_\natural x| \le \iota \). Setting \(t = \alpha (|T_\natural ^\perp x|/\iota )\) we obtain
Proof of (f): Let \(L \in {\mathbf {K}}_*\) be such that \({\mathbf {l}}(L) \ge \frac{1}{2} {\mathbf {l}}(K)\) and \(L \cap {\text {Int}}_{\mathrm {c}}(K) = \varnothing \). Observe that if \({\mathbf {l}}(L) > \frac{1}{2} {\mathbf {l}}(K)\), then L is a sum of some cubes from \({\mathbf {K}}_*\) which do not intersect \({\text {Int}}_{\mathrm {c}}(K)\) and have side length \(\frac{1}{2} {\mathbf {l}}(K)\); thus, it is enough to prove the claim in case \({\mathbf {l}}(L) = \frac{1}{2} {\mathbf {l}}(K)\)–we shall assume this holds. Since \(\varphi (x) = x\) for \(x \in {\mathbf {R}}^n\) with \({{\mathrm{dist}}}(x,K) \ge \varepsilon \) and \(\varepsilon \le \frac{1}{4} {\mathbf {l}}(K)\) we will also assume that \(L \cap K \ne \varnothing \). For \(\kappa \in \{-1,0,1\}^n\) and \(\lambda \in \{-2,-1,1,2\}^n\) let \(c_\kappa \), \(T_{\kappa }\), \(R_{\lambda }\), be as in 5.1. If \(\dim ({T}_\natural [L ]) = \dim (K)\), then \({T}_\natural [L ]= T \cap r [R_\lambda ]\) for some \(\lambda \in \{-2,-1,1,2\}^n\). If \(\dim ({T}_\natural [L ]) < \dim (K)\), then \({T}_\natural [L ]\subseteq {\text {Bdry}}_{\mathrm {c}}(K)\) and L is contained in some face of R. In this case let \(\kappa \in \{-1,0,1\}^n\) be such that \(L \subseteq r [F_{\kappa } ]\) and \(F_{\kappa } \subseteq F_{\sigma }\) whenever \(L \subseteq r [F_{\sigma } ]\) for some \(\sigma \in \{-1,0,1\}^n\). If it happens that \(\dim (F_\kappa ) > \dim (L)\), then L must lie inside \(T_\sigma \) for some \(\sigma \in \{-1,0,1\}^n\). Altogether, there exist \(\lambda \in \{-2,-1,1,2\}^n {{\mathrm{\sim }}}\{-2,2\}^n\) and \(\kappa , \sigma \in \{-1,0,1\}^n\) such that
Hence, \(\psi [{T}_\natural [L ]]\subseteq {T}_\natural [L ]\) by 7.9(d). Since L is convex, we obtain \(\varphi [{T}_\natural [L ]]\subseteq {T}_\natural [L ]\). Finally, note that \(L = {T}_\natural [L ]+ T_\natural ^\perp [L ]\) and \(\varphi [L ]= \varphi [{T}_\natural [L ]]+ T_\natural ^\perp [L ]\), which proves the claim in case \(L \cap {\text {Int}}_{\mathrm {c}}(K) = \varnothing \).
If \({\text {Int}}_{\mathrm {c}}(K) \cap L \ne \varnothing \) but \({\mathbf {l}}(L) \ge {\mathbf {l}}(K)\), then \(K \subseteq {T}_\natural [L ]\). Clearly \(\varphi [K ]\subseteq K\) and \({T}_\natural [L ]{{\mathrm{\sim }}}K\) is contained in a sum of cubes with side length at least \(\frac{1}{2} {\mathbf {l}}(K)\) which do not intersect \({\text {Int}}_{\mathrm {c}}(K)\). Hence, \(\varphi [{T}_\natural [L ]]\subseteq {T}_\natural [L ]\) and the claim follows as before.
Proof of (g): Assume \(x \in {\mathbf {R}}^n\) satisfies \({{\mathrm{dist}}}(x, {\text {Bdry}}_{\mathrm {c}}(K)) \le \varepsilon \), then \({{\mathrm{dist}}}({T}_\natural x, {\text {Bdry}}_{\mathrm {c}}(K)) \le \varepsilon \). Since \(d \le \iota /2 \le 2^{-7} {{\mathrm{dist}}}(a, {\text {Bdry}}_{\mathrm {c}}(K))\) we see that \(|{T}_\natural x - a| \ge (1-2^{-8}) {{\mathrm{dist}}}(a, {\text {Bdry}}_{\mathrm {c}}(K)) \ge d\) so \(\psi ({T}_\natural x) = \psi _a({T}_\natural x)\). Recalling \(\alpha '(t) \le 2\), \(\alpha (t) \le 1\) for \(t \in {\mathbf {R}}\), \(\iota = \varepsilon /\sqrt{2}\), and 7.9 (f, g) we get
Proof of (h): Let us assume \(\max \{m_1,\ldots ,m_l\} \le k-1\); hence, \(r(a) \in \bigl [-\frac{1}{2}, \frac{1}{2} \bigr ]^n \cap T\). Note that for \(i \in \{1,\ldots ,l\}\) and \(x \in T {{\mathrm{\sim }}}{\mathbf {B}}(a,d)\)
Using (33) and the definition of \(\Sigma \), U, and \(\mu _i\) we get
Combining (32) with (34) and (35) yields (h). \(\square \)
Next, we shall prove our main deformation Theorem 7.13. Given a finite subset \({\mathcal {A}}\) of an admissible family \({\mathcal {F}}\) of top dimensional cubes from \({\mathbf {K}}\) and some sets \(\Sigma _1\), ..., \(\Sigma _l\), we deform all these sets onto the m dimensional skeleton of \({\mathcal {A}}\) using a smooth deformation of \({\mathbf {R}}^n\). Furthermore, we provide estimates on the measure of the deformed sets (i.e. the images of \(\Sigma _i\) for \(i =1,2,\ldots ,l\)) and, in case \(\Sigma _i\) are rectifiable, also on the measure of the whole deformation (i.e. the images of \([0,1] \times \Sigma _i\) for \(i =1,2,\ldots ,l\)). The basic idea of the proof is simple: we order all the cubes of the cubical complex \(\mathbf {CX}({\mathcal {F}})\) which touch the interior of some cube from \({\mathcal {A}}\) lexicographically with respect to side length and dimension and then apply 7.11 iteratively to each cube. If the dimensions of \(\Sigma _1\), ..., \(\Sigma _l\) all equal m, then we additionally ensure that all the m-dimensional faces of \({\mathcal {A}}\) are either fully covered or not covered at all. During this last step we cannot control the derivative so we actually provide two deformations: one with good estimates (called “g”) and one without estimates (called “f”) but performing the last step of cleaning the cubes which are not fully covered.
To be able to estimate the measure of the image of \(\Sigma _i\) even if \(\Sigma _i\) is not rectifiable, we need the following simple lemma.
Lemma 7.12
Let \(S \subseteq {\mathbf {R}}^n\) be such that \({\mathscr {H}}^m(S \cap K) < \infty \) for every compact set \(K \subseteq {\mathbf {R}}^n\), \(g \in {\mathscr {C}}^1({\mathbf {R}}^n,{\mathbf {R}}^N)\) for some \(N \in {\mathscr {P}}\), and 
be non-negative. Then
Proof
Since S can be decomposed into a countable sum of compact sets we can assume S is compact. Furthermore, using standard methods of Lebesgue integration [19, 2.3.3, 2.4.8] we can assume \(f = \mathbb {1}_{A}\) for some 
measurable set \(A \subseteq {\mathbf {R}}^N\). Let \(\varepsilon > 0\). For each \(x \in S\) choose \(r_x > 0\) so that
This is possible since \(\mathrm {D}g\) is continuous. From the family \(\{ {\mathbf {U}}(x,r_x) : x \in S \}\) choose a finite covering \({\mathcal {B}} = \{ B_1, \ldots , B_K \}\) of S. For \(j = 1,2,\ldots ,K\) define \(S_j \subseteq S\), and \(x_j \in {\mathbf {R}}^n\), and \(r_j \in {\mathbf {R}}\) so that
We obtain
where
We obtain the claim by letting \(\varepsilon \rightarrow 0\) and using the dominated convergence theorem; see [19, 2.4.9]. \(\square \)
Theorem 7.13
Suppose
Then for each \(\varepsilon \in (0,\varepsilon _0)\), setting \(G_{\varepsilon } = \mathop {{\textstyle \bigcup }}{\mathcal {A}} + {\mathbf {U}}(0,\varepsilon )\) and \(G_0 = {{\mathrm{Int}}}\mathop {{\textstyle \bigcup }}{\mathcal {A}}\), there exist deformations \(f,g \in {\mathscr {C}}^{\infty }({\mathbf {R}}\times {\mathbf {R}}^n, {\mathbf {R}}^n\)) and a neighbourhood U of \(\Sigma \cap G_{\varepsilon }\) in \({\mathbf {R}}^n\) satisfying:
-
(a)
\(f(t,x) = x\) if either \(t = 0\) and \(x \in {\mathbf {R}}^n\) or \(t \in {\mathbf {R}}\) and \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}G_{\varepsilon }\).
-
(b)
There exist \(N,N_0 \in {\mathscr {P}}\), \(N_0 \le N\), \(\varphi _1, \ldots , \varphi _N \in {\mathfrak {D}}({G_{\varepsilon }})\), and \(K_1,\ldots ,K_N \in \mathbf {CX}({\mathcal {F}})\) such that for each \(j = 1,\ldots ,N\) setting \(\psi _0 = \mathrm {id}_{{\mathbf {R}}^n}\) and \(\psi _j = \varphi _j \circ \psi _{j-1}\) we have
$$\begin{aligned} \bigl \{ x \in {\mathbf {R}}^n : \varphi _j(x) \ne x \bigr \} \subseteq K_j + {\mathbf {U}}(0,\varepsilon ) \quad \text {and} \quad \varphi _j [\psi _{j-1} [U ]\cap K_j ]\subseteq {\text {Bdry}}_{\mathrm {c}}(K_j). \end{aligned}$$Moreover, there exists \(s \in {\mathscr {C}}^{\infty }({\mathbf {R}}, {\mathbf {R}})\) such that \(s(0) = 0\), and \(s(1) = 1\), and \(0 \le s'(t) \le 2\) for \(t \in {\mathbf {R}}\), and \(\mathrm {D}^{k}s(0) = 0 = \mathrm {D}^{k}s(1)\) for \(k \in {\mathscr {P}}\), and
$$\begin{aligned} f(t,\cdot ) = s(tN-j) \psi _{j+1} + (1 - s(tN-j)) \psi _{j} = g(tN/N_0, x) \end{aligned}$$whenever \(j \in \{0,\ldots ,N-1\}\) and \(t \in {\mathbf {R}}\) satisfy \(j \le tN \le j+1\).
-
(c)
\(f(t,\cdot ) \in {\mathfrak {D}}({G_{\varepsilon }})\) for each \(t \in I\).
-
(d)
If \(K \in \mathbf {CX}({\mathcal {F}})\), then \(f(t,\cdot )[K ]\subseteq K\) for \(t \in I\). In particular
$$\begin{aligned} f(t,\cdot )\left[ \mathop {{\textstyle \bigcup }}{\mathcal {A}} \right] \subseteq \mathop {{\textstyle \bigcup }}{\mathcal {A}} \quad \text {and} \quad f(t,\cdot )\left[ {\mathbf {R}}^n {{\mathrm{\sim }}}\mathop {{\textstyle \bigcup }}{\mathcal {A}} \right] \subseteq {\mathbf {R}}^n {{\mathrm{\sim }}}G_0 \quad \hbox { for}\ t \in I. \end{aligned}$$ -
(e)
\(g(1,\cdot )[U ]\cap G_0 \subseteq \mathop {{\textstyle \bigcup }}\bigl \{ K \in \mathbf {CX}({\mathcal {F}}) : K \cap G_0 \ne \varnothing ,\, \dim (K) = m \bigr \}\).
-
(f)
If \(m_1 = \cdots = m_l\), then for each \(K \in \mathbf {CX}({\mathcal {F}}) \cap {\mathbf {K}}_{m}\) satisfying \(K \cap G_0 \ne \varnothing \) there holds
$$\begin{aligned} \text {either} \quad {\text {Int}}_{\mathrm {c}}(K) \cap f(1,\cdot )\left[ \Sigma \cap \mathop {{\textstyle \bigcup }}{\mathcal {A}}\right] = \varnothing \quad \text {or} \quad K \cap f(1,\cdot )\left[ \Sigma \cap \mathop {{\textstyle \bigcup }}{\mathcal {A}}\right] = K. \end{aligned}$$ -
(g)
There exists \(\Gamma = \Gamma (n,m) \in (0,\infty )\) such that for each \(t \in I\) and \(i \in \{1,\ldots ,l\}\) and \(Q \in {\mathcal {F}}\), setting \({{\widetilde{Q}}} = \mathop {{\textstyle \bigcup }}\{ R \in {\mathcal {F}} : R \cap Q \ne \varnothing \}\),
$$\begin{aligned}&\Vert \mathrm {D}f(t,\cdot )(x) \Vert < \Gamma \quad \hbox { for}\ x \in G_{\varepsilon } {{\mathrm{\sim }}}G_0, \end{aligned}$$(36)$$\begin{aligned}&\quad \int _{\Sigma _i \cap Q} \Vert \mathrm {D}g(t,\cdot )\Vert ^{m_i} \,\mathrm {d}{\mathscr {H}}^{m_i} < \Gamma {\mathscr {H}}^{m_i}\bigl ( \Sigma _i \cap ({{\widetilde{Q}}} + {\mathbf {U}}(0,\varepsilon )) \bigr ), \end{aligned}$$(37)$$\begin{aligned}&\quad {\mathscr {H}}^{m_i}(g(t,\cdot ) [\Sigma _i \cap G_{\varepsilon } ]) \mathop {\le }\limits ^{7.12} \int _{\Sigma _i \cap G_{\varepsilon }} \Vert \mathrm {D}g(t,\cdot )\Vert ^{m_i} \,\mathrm {d}{\mathscr {H}}^{m_i} < \Gamma {\mathscr {H}}^{m_i}(\Sigma _i \cap G_{\varepsilon }), \end{aligned}$$(38)$$\begin{aligned}&\quad {\mathscr {H}}^{m_i}(f(1,\cdot ) [\Sigma _i ]\cap Q) < \Gamma {\mathscr {H}}^{m_i}\bigl ( \Sigma _i \cap ({{\widetilde{Q}}} + {\mathbf {U}}(0,\varepsilon )) \bigr ), \end{aligned}$$(39)$$\begin{aligned}&\quad {\mathscr {H}}^{m_i}(f(1,\cdot ) [\Sigma _i \cap G_{\varepsilon } ]) < \Gamma {\mathscr {H}}^{m_i}(\Sigma _i \cap G_{\varepsilon }); \end{aligned}$$(40)moreover, if \(\Sigma _i\) is \(({\mathscr {H}}^{m_i},m_i)\) rectifiable, then
$$\begin{aligned} {\mathscr {H}}^{m_i+1}(g [I \times (\Sigma _i \cap G_{\varepsilon }) ]) \le {\mathscr {H}}^{m_i+1}(f [I \times (\Sigma _i \cap G_{\varepsilon }) ]) < \Gamma \delta {\mathscr {H}}^{m_i}(\Sigma _i \cap G_{\varepsilon }).\nonumber \\ \end{aligned}$$(41)
Proof
Fix \(\varepsilon \in (0,\varepsilon _0)\). Without loss of generality we assume that \(\Sigma _i \subseteq G_{\varepsilon }\) for \(i = 1,2,\ldots ,l\). Define
Let \(\Delta = \Delta (n,m) \in {\mathscr {P}}\) be so big that
Let \(\{ K_1,\ldots ,K_{N_0} \} = {\mathcal {C}}\) be an enumeration of \(\mathcal C\) chosen so that for \(1 \le i \le j \le N_0\)
For each \(j = 0,1,\ldots ,N_0\) we shall define inductively maps \(\varphi _j\), \(\psi _j\), \(\zeta _{Q,j}\), \(\eta _{Q,j}\), measures \(\nu _{Q,j,i}\), and open sets \(U_j \subseteq {\mathbf {R}}^n\), where \(i \in \{ 1,2,\ldots ,l \}\) and \(Q \in {\mathcal {F}}\). First we set
Assume that \(\varphi _{j-1}\), \(\psi _{j-1}\), \(\eta _{Q,j-1}\), \(U_{j-1}\), \(\nu _{j-1,1}, \ldots , \nu _{j-1,l}\) are defined for some \(j = 1,2,\ldots ,N_0\). We set
Observe, that all the measures \(\nu _{Q,i,j}\) for \(i \in \{ 1,2,\ldots ,l \}\) and \(Q \in {\mathcal {F}}\) are Radon because \(\eta _{Q,j-1}\) is smooth and proper and 
is finite. Let \(\varphi _j\) and \(U_j\) be the map and the neighbourhood of \(\Sigma \) constructed by employing 7.11 with
If \(Q \cap K_j \ne \varnothing \), we set \(\zeta _{Q,j} = \varphi _{j}\), and if \(Q \cap K_j = \varnothing \), we set \(\zeta _{Q,j} = \mathrm {id}_{{\mathbf {R}}^n}\). Next, we define
If \(m_1 > m_l\), then we set \(N = N_0\). If \(m = m_1 = \cdots = m_l\), then we still have to take care of the cubes of dimension m which are not fully covered. In this case we define
We enumerate \({\mathcal {C}}' = \{ K_{N_0+1}, \ldots , K_{N} \}\) so that for \(N_0 < i \le j \le N\) we have \({\mathbf {l}}(K_i) \ge {\mathbf {l}}(K_j)\). For \(j = N_0+1, \ldots , N\) we define inductively \(\varphi _j\), \(\psi _j\), \(U_j\) similarly as before by employing 7.11 with \(K_j\), \(\varepsilon /\Delta \), l, 
, ..., 
, m, ..., m in place of K, \(\varepsilon _0\), l, \(\nu _1\), ..., \(\nu _l\), \(m_1\), ..., \(m_l\) and we set \(\psi _j = \varphi _j \circ \psi _{j-i}\).
Let \(f(t,\cdot )\) for \(t \in I\) be defined as in (b). For \(t > 1\) we set \(f(t,\cdot ) = f(1,\cdot )\) and for \(t < 0\) we set \(f(t,\cdot ) = f(0,\cdot )\). Then we define \(g(t,\cdot ) = f(tN_0/N,\cdot )\) for all \(t \in {\mathbf {R}}\) and we set \(U = \mathop {{\textstyle \bigcap }}\bigl \{ U_j : j = 1,2,\ldots , N \bigr \}\). Clearly (a) and (b) are satisfied.
Proof of (c): First observe that \(\varphi _{j} \in {\mathfrak {D}}({K_j + {\mathbf {U}}(0,\varepsilon /\Delta )})\) for each \(j = 1,2,\ldots ,N\) due to 7.11(a). Since \(K_j + {\mathbf {B}}(0,\varepsilon /\Delta ) \subseteq G_{\varepsilon }\) we see that \(\psi _j \in {\mathfrak {D}}({G_{\varepsilon }})\).
Fix \(t \in I\) and choose \(j \in {\mathscr {P}}\) such that \(j \le tN \le j+1\). We have
Since \(\varphi _{j+1} \in {\mathfrak {D}}({K_j + {\mathbf {U}}(0,\varepsilon /\Delta )})\) and \(K_{j+1} + {\mathbf {U}}(0,\varepsilon /\Delta )\) is convex we see that
hence, \(f(t,\cdot ) \in {\mathfrak {D}}({G_{\varepsilon }})\).
Proof of (d): Since \({\mathcal {F}}\) is admissible for \(j \in \{1,2,\ldots ,N\}\) and \(L \in \mathbf {CX}({\mathcal {F}})\) exactly one of the following options holds
-
\(L \cap K_j = \varnothing \) and then \(\varphi _j[L ]= L\) by 7.11(b);
-
\(L \cap K_j \ne \varnothing \) and \({\mathbf {l}}(L) \ge \frac{1}{2} {\mathbf {l}}(K_j)\) and \(L \cap {\text {Int}}_{\mathrm {c}}(K_j) = \varnothing \) and \(\varphi _j[L ]\subseteq L\) by 7.11(f);
-
\(L \cap K_j \ne \varnothing \) and \(L \cap {\text {Int}}_{\mathrm {c}}(K_j) \ne \varnothing \) and \(\dim (L) > \dim (K_j)\) and \({\mathbf {l}}(L) \ge {\mathbf {l}}(K_j)\), because \(K_j \in \mathbf {CX}({\mathcal {F}})\), and \(\varphi _j[L ]\subseteq L\) by 7.7 and 7.11 (f).
In consequence, if \(L \in \mathbf {CX}({\mathcal {F}})\), then \(\psi _j[L ]\subseteq L\) for \(j \in \{1,\ldots ,N\}\) and, since L is convex, we obtain \(f(t,\cdot ) [L ]\subseteq L\) for \(t \in I\). In particular
Proof of (e): For \(j \in \{1,\ldots ,N\}\) set
For \(j = 1, \ldots , N\) set
We shall prove by induction the following claim:
If \(j=1\), then \(k = n\) and \(\psi _1 = \varphi _1\) and claim (43) follows from 7.11(c, f).
Assume \(j \in \{2,3,\ldots ,N_0\}\) and \(k = \dim (K_j)\). By inductive hypothesis, if \(j > \alpha (j)\), then
and if \(j = \alpha (j)\), then \(j-1 = \beta (j-1)\) and
Claim (43) follows now by the following observations:
-
if \(j > \alpha (j)\) and \(Q \in {\mathcal {C}}_{j-1} \cap {\mathbf {K}}_{k-1}\), then \(Q \cap {\text {Int}}_{\mathrm {c}}(K_j) = \varnothing \) so \(\varphi _j[Q ]\subseteq Q\) by (42);
-
if \(\alpha (k) \le j \le \beta (j)\), we have \(\varphi _j [\psi _{j-1} [U ]\cap K_j ]\subseteq {\text {Bdry}}_{\mathrm {c}}(K_j) \subseteq \mathop {{\textstyle \bigcup }}\bigl ( {\mathcal {C}}_j \cap {\mathbf {K}}_{k-1}\bigr )\) by 7.11(c);
-
if \(\alpha (k) \le j \le \beta (j)\) and \(Q \in ({\mathcal {C}}_{\beta (j)} {{\mathrm{\sim }}}{\mathcal {C}}_j) \cap {\mathbf {K}}_k\), then \(\varphi _j[Q ]\subseteq Q\) by (42).
Proof of (f): This follows by (e) and the construction.
Proof of (g): To prove (36) take \(x \in G_{\varepsilon } {{\mathrm{\sim }}}G_0\) and note that are at most \(\Delta \) maps amongst \(\varphi _1\), ..., \(\varphi _N\) which move the point x so, recalling 7.11(g), we get
To prove (37) choose a cube \(Q \in {\mathcal {F}}\) and \(i \in \{1,2,\ldots ,l\}\). Set
Observe that for \(j \in \{ 1,2,\ldots , N_0 \}\) we have
Therefore, we may write
If \(\zeta _{Q,j} = \mathrm {id}_{{\mathbf {R}}^n}\), then we obtain
If \(\zeta _{Q,j} = \varphi _k\) for some \(k \in \{1,2,\ldots ,N_0\}\), then \(K_k \subseteq {{\widetilde{Q}}}_{\varepsilon }\) and if \(K_k\) is a face of some cube \(R \in {\mathbf {K}}_*\) with \(\dim R > \dim K_k\), then \({{\mathrm{spt}}}\nu _{Q,j} \cap {\text {Int}}_{\mathrm {c}}(R) = \varnothing \); thus, employing 7.11(b)(g)(h), we get
Now, the second case (when \(\zeta _{Q.j} = \varphi _k\) for some \(k \in \{1,2,\ldots ,N_0\}\)) can happen at most \(\Delta \) times so combining (47), (46), and (45)
We may now finish the proof of (37) by writing
whenever \(j \in \{1,2,\ldots ,N_0\}\) and \(t \in {\mathbf {R}}\) are such that \(j \le N_0t \le j+1\).
To prove (38) we write
To prove (39) and (40) note that for \(Q \in {\mathcal {F}}\), by construction (in particular by (f)),
and, recalling (d) and (44),
Assume now that \(\Sigma _i\) is \(({\mathscr {H}}^{m_i},m_i)\) rectifiable. Then \(I \times \Sigma _i\) is \(({\mathscr {H}}^{m_i+1},m_i+1)\) rectifiable by [19, 3.2.23] and we may use the area formula [19, 3.2.20] to prove (41). Define
Let \(\tau = (1,0) \in {\mathbf {R}}\times {\mathbf {R}}^n\) be the “time direction” and set \(g_t(x) = g(t,x)\) for \((t,x) \in {\mathbf {R}}\times {\mathbf {R}}^n\). Observe that for \((t,x) \in I \times \Sigma \) and \(j \in {\mathscr {P}}\) such that \(j \le tN_0 \le j+1\) we have
hence, using [19, 3.2.20, 3.2.23], (b), (d), and 7.11(b)(e)
If \(m = m_1 > m_l\), then \(N = N_0\) and \(f = g\) and there is nothing more to prove. Otherwise, we have \(m = m_1 = \cdots = m_l\) and \(g [I \times (\Sigma _i \cap G_{\varepsilon }) ]= f [I' \times (\Sigma _i \cap G_{\varepsilon }) ]\), where \(I' = [0,N_0/N]\). Hence, we need to estimate \({\mathscr {H}}^{m+1}(f [(I {{\mathrm{\sim }}}I') \times (\Sigma _i \cap G_{\varepsilon }) ])\). Observe, that
so \({\mathscr {H}}^{m+1}(f [(I {{\mathrm{\sim }}}I') \times (\Sigma _i \cap G_{0}) ]) = 0\). On the other hand \(\Vert \mathrm {D}f_t(x) \Vert \le \Gamma _{{7.11}}\) for \(x \in G_{\varepsilon } {{\mathrm{\sim }}}G_0\) and \(t \in I {{\mathrm{\sim }}}I'\) so \({\mathscr {H}}^{m+1}(f [(I {{\mathrm{\sim }}}I') \times (\Sigma _i \cap G_{\varepsilon } {{\mathrm{\sim }}}G_{0}) ])\) can be estimated as in (48). \(\square \)
We finish this section with a small lemma that allows to apply 4.3 to the mapping constructed in 7.13.
Lemma 7.14
If \(f \in {\mathscr {C}}^{1}({\mathbf {R}}^n,{\mathbf {R}}^n)\) and \(U \subseteq {\mathbf {R}}^n\) and \(\dim _{{\mathscr {H}}}(f [U ]) \le m\), then \(\dim {{\mathrm{im}}}\mathrm {D}f(x) \le m\) for \(x \in U\).
Proof
Assume there exists a point \(x \in U\) such that \(\dim {{\mathrm{im}}}\mathrm {D}f(x) = k>m\). Define \(L = {{\mathrm{im}}}\mathrm {D}f(x) \in {\mathbf {G}}(n,k)\) and set \(g = {L}_\natural \circ f\). Observe that
Moreover, since \(f(y) = f(x) + \mathrm {D}f(x)(y-x) + o(|x-y|)\) we see that for small enough \(r > 0\) we have
Hence, for some \(r > 0\) we obtain \({\mathscr {H}}^k(f [{\mathbf {B}}(x,r) ]) > 0\) which contradicts \(\dim _{{\mathscr {H}}}(f [U ]) \le m\). \(\square \)
8 Slicing varifolds by continuously differentiable functions
We recall the theory developed by Almgren in [4, I.3] and [2, §7].
In this sections we shall always assume \(U \subseteq {\mathbf {R}}^n\) is open, \(m,n,\nu \in {\mathbf {Z}}\) are such that \(0 \le \nu \le m < n\), \(V \in {\mathbf {V}}_{m}(U)\), \(f \in {\mathscr {C}}^1(U,{\mathbf {R}}^{\nu })\) is proper, and \(\pi : {\mathbf {R}}^n \times {\mathbf {G}}(n,m) \rightarrow {\mathbf {R}}^n\) is the projection onto the first factor.
Definition 8.1
Whenever \(\beta \in {\mathscr {K}}(U \times {\mathbf {G}}(n,m-\nu ))\) and \(\varphi \in {\mathscr {K}}({\mathbf {R}}^{\nu })\) we set
It was shown in [4, I.3(2)] that \((V,f) \in {\mathbf {V}}_{{}m - \nu }(U)\) and \(\mu _{\beta }\) is a Radon measure for each \(\beta \in {\mathscr {K}}(U \times {\mathbf {G}}(n,m-\nu ))\).
Definition 8.2
The slice of V with respect to f at \(t \in {\mathbf {R}}^{\nu }\) is the varifold \(\langle V, f, t \rangle \in {\mathbf {V}}_{{}m - \nu }(U)\), satisfying
Remark 8.3
By [4, I.3(2)], there exists \(\langle V, f, t \rangle \in {\mathbf {V}}_{{}m - \nu }(U)\) and, since f is proper, \({{\mathrm{spt}}}\Vert \langle V, f, t \rangle \Vert \) is compact for \({\mathscr {L}}^{\nu }\) almost all \(t \in {\mathbf {R}}^{\nu }\). Next, we view \(\{ V \in {\mathbf {V}}_{{}m-\nu }(U) : {{\mathrm{spt}}}\Vert V\Vert \text { is compact} \}\) as a subset of the vectorspace (cf. [19, 2.5.19])
Then \(\{ W \in {\mathscr {K}}(U \times {\mathbf {G}}(n,m - \nu ))^{*} : {{\mathrm{spt}}}W \text { is compact} \}\) becomes a separable normed vectorspace such that the the norm topology coincides with the weak topology on subspaces of W supported in a fixed compact set.
Definition 8.4
(cf. [4, I.3(3)]) The Lebesgue set of the slicing operator \(\langle V, f, \cdot \rangle \) is the set of those \(t \in {\mathbf {R}}^{\nu }\) for which
Remark 8.5
Note that \({\mathscr {L}}^\nu \) almost all \(t \in {\mathbf {R}}^\nu \) are Lebesgue points of \(\langle V, f, \cdot \rangle \); see [4, I.3(3)] for the proof.
Remark 8.6
Recalling [4, I.3(4)] we see that if \(S \subseteq U\) is \(({\mathscr {H}}^m,m)\) rectifiable and \({\mathscr {H}}^m\) measurable and bounded, then
Next, we define the product of a varifold with a cube as in [4, I.3(5)]. Products of more general varifolds were described also in [29, §3].
Definition 8.7
If \(l \in {\mathbf {Z}}\), and \(l \ge 1\), and \(I = \{ t \in {\mathbf {R}}: 0 \le t \le 1\}\), and \(j_l : {\mathbf {R}}^l \rightarrow {\mathbf {R}}^l \times {\mathbf {R}}^n\) and \(j_n : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^l \times {\mathbf {R}}^n\) are injections, then
for \(\alpha \in {\mathscr {K}}({\mathbf {R}}^{l} \times U \times {\mathbf {G}}(l+n,l+m))\).
Definition 8.8
For \(t \in {\mathbf {R}}\) and \(\delta \in (0,1)\) and \(\rho : {\mathbf {R}}^n \rightarrow {\mathbf {R}}\) we define the functions
by requiring that \(s_{\delta }\) is of class \({\mathscr {C}}^{\infty }\) and
Lemma 8.9
Let \(V \in {\mathbf {V}}_{m}(U)\) and \(\rho \in {\mathscr {C}}^1(U,{\mathbf {R}})\) be a proper map, and \(\iota \in (0,\infty )\), and \(t \in {\mathbf {R}}\) be a Lebesgue point of \(\langle V, \rho , \cdot \rangle \) such that 
and
Set 
and 
. Then
Proof
Since \(\rho \) and t are fixed we abbreviate \(K_{\delta } = K_{\rho ,t,\delta }\). For \(\delta \in (0,1)\) define
Clearly
and \(\lim _{\delta \downarrow 0} i_{1\,\#} V_{1,\delta } = i_{1\,\#} V_{1}\) so it suffices to prove that \(\lim _{\delta \downarrow 0} K_{\delta \,\#} V_{2,\delta } = {\mathbf {v}}_{1}(I) \times \langle V, \rho , t \rangle \). To this end it is enough to show that \(\lim _{\delta \downarrow 0} \vert \vert \vert K_{\delta \,\#} V_{2,\delta } - {\mathbf {v}}_{1}(I) \times \langle V, \rho , t \rangle \vert \vert \vert = 0\).
Let \(j_i : {\mathbf {R}}\rightarrow {\mathbf {R}}\times U\) and \(j_n : U \rightarrow {\mathbf {R}}\times U\) be injections and let \(\pi : U \times {\mathbf {G}}(n,m-1) \rightarrow U\) be the projection onto the first factor. For \(\Vert V\Vert \) almost all x we define T, \(R_{\delta }\), P, \(J_{K,\delta }\), and \(J_{\rho }\) by requiring
Whenever \(x \in {{\mathrm{dmn}}}T\) and \(Q \in {\mathbf {G}}(n,m-1)\) and \(\tau \in [0,1]\) we also set
Let \(\varepsilon \in (0,1/2)\). If \(\vert \vert \vert \langle V, \rho , t \rangle \vert \vert \vert > 0\), then assume additionally that \(\varepsilon \le 2^{-5} \vert \vert \vert \langle V, \rho , t \rangle \vert \vert \vert ^{-1}\). Find \(\delta _0 \in (0,1)\) such that for all \(\delta \in (0,\delta _0)\)
Such \(\delta _0 > 0\) exists because t is a Lebesgue point of \(\langle V, \rho , \cdot \rangle \) and we assumed (49). It follows from (51), applied to \([t-\delta ^2,t]\) and \([t-\delta +\delta ^2,t]\) and \([t-\delta ,t]\), and from (50) that
For any \(\alpha \in {\mathscr {K}}({\mathbf {R}}\times U \times {\mathbf {G}}(n+1,m))\) such that \(\sup {{\mathrm{im}}}|\alpha | \le 1\) and \({{\mathrm{Lip}}}\alpha \le 1\), employing the co-area formula [19, 3.2.22], we get
Since \({{\mathrm{Lip}}}\gamma _{\tau } = 1\) for \(\tau \in [0,1]\), we have by (51)
To estimate \(B_1(\alpha ,\delta )\) we set
Clearly 
for \(s \in [t-\delta ,t]\). We estimate first the third part. Straightforward computations (cf. [4, I.3(1)]) show that if \((x,S) \in X\) and \(\tau \in [0,1]\) and \(\delta \in (0,1)\) and \(\rho (x) = t - \delta \tau \), then, setting \(v = {T(x)}_\natural ({{\mathrm{grad}}}\rho (x)) \in S^{\perp } \cap T\),
Hence, recalling 8.8 we see that \((t - \rho (x))/\delta \in [0, \delta ) \cup (1-\delta ,1]\) whenever \((x,S) \in A_3(\delta )\) so
For \((x,S) \in A_1(\delta )\) we have \(J_{K,\delta }(x) \le (1 + \varepsilon ^{-4} \delta ^2)^{1/2} \le 1 + \varepsilon ^{-4}\) and \(\delta J_{K,\delta }(x) / J_{\rho }(x) \ge 1\) so, using the co-area formula [19, 3.2.22] and \(\sup {{\mathrm{im}}}|\alpha | \le 1\), we obtain
To deal with \(A_2(\delta )\) first observe that \(\delta J_{K,\delta }(x)/J_{\rho }(x) \ge 1\) and \(J_{\rho }(x) > \delta \varepsilon ^{-2}\) and \(\varepsilon \le 1/2\) imply
where \(\tau = (t - \rho (x))/\delta \). Therefore, by (57) and (56) and (50),
If \((x,S) \in A_2(\delta )\), then \(\delta \le \varepsilon ^2 J_{\rho }(x)\) so \(\delta J_{K,\delta }(x) / J_{\rho }(x) \le 1 + \varepsilon ^4\) and, using \({{\mathrm{Lip}}}\alpha \le 1\),
Recall that \(\varepsilon < 1/2\) and if \(\vert \vert \vert \langle V, \rho , t \rangle \vert \vert \vert > 0\), we assumed \(\varepsilon \le 2^{-5} \vert \vert \vert \langle V, \rho , t \rangle \vert \vert \vert ^{-1}\). Thus, using (51)
Finally, combining (54), (55), (58), (59), (60), (61) we see that \(|W_{\delta }(\alpha )| \le \varepsilon \) for \(\delta \in (0,\delta _0)\). Since \(\delta _0\) was chosen independently of \(\alpha \) we have \(\vert \vert \vert W_{\delta } \vert \vert \vert \le \varepsilon \) for \(\delta \in (0,\delta _0)\). \(\square \)
Corollary 8.10
Assume \(a,b \in {\mathbf {R}}\) are such that 
. Then for \({\mathscr {L}}^1\) almost all \(t \in (a,b)\)
9 Density ratio bounds
The main result 9.3 of this section gives lower and upper bounds on the density ratios of \(\Vert V\Vert \) for any V which minimises a bounded \({\mathscr {C}}^0\) integrand F (not necessarily elliptic). Our proof follows the ideas presented in [3, 2.9(b2)(b3), 3.2(a)(b), 3.4(2) last paragraphs on pp. 347 and 348] as well as in [21, 7.8, 8.2].
Let \(a \in U \subseteq {\mathbf {R}}^n\) be fixed, \(\rho (x) = |x-a|\), and \(M(r) = {\Vert V\Vert }\,{{\mathbf {U}}(a,r)}\) for \(r > 0\). The key point is to prove the differential inequalities (78) and (80). Assume \(M'(r_0)\) exists and is finite for some \(r_0 > 0\)–this holds for \({\mathscr {L}}^1\) almost all \(r_0 \in (0,\infty )\). Recall that V is a limit of some sequence of the form \(\{ {\mathbf {v}}_{m}(S_i \cap U) : i \in {\mathscr {P}}\}\), where \(S_i\) belong to a good class. Our strategy is to first choose a competitor \(S \subseteq {\mathbf {R}}^n\) such that \({\mathbf {v}}_{m}(S \cap U)\) is weakly close to V, then construct an admissible deformation D (using the deformation Theorem 7.13) of S such that the \({\mathscr {H}}^m(D[S ])\) can be estimated in terms of \(M'(r_0)\), and finally use minimality of V to derive estimates on \(M(r_0)\).
More precisely we proceed as follows. We choose a competitor \(S \subseteq {\mathbf {R}}^n\) so that the 4d-neighbourhood of \(S \cap \rho ^{-1}\{r_0\}\) has \({\mathscr {H}}^m\) measure controlled roughly by \(d M'(r_0)\), where \(d > 0\) is a small number. This is possible because the mass function \(V \mapsto \Vert V\Vert ({\mathbf {R}}^n)\) is continuous on the space of varifolds supported in a fixed compact set. Then, we use the deformation Theorem 7.13 to “project” the part of S lying inside 2d-neighbourhood of \(\rho ^{-1}\{r_0\}\) onto some m dimensional cubical complex and we denote the deformed set R. After this step the part of R lying in a 2d-neighbourhood of \(\rho ^{-1}\{r_0\}\) is \(({\mathscr {H}}^m,m)\) rectifiable (as a finite sum of m dimensional cubes) and, moreover, its measure is still controlled by \(d M'(r_0)\) due to the first part of 7.13(g) which holds for non-rectifiable sets. Next, we use the co-area formula (valid on rectifiable sets) together with the Chebyshev inequality and 8.5 to find some \(r_1 > 0\) in the d-neighbourhood of \(r_0\) so that the \({\mathscr {H}}^{m-1}\) measure of the slice \(R \cap \rho ^{-1}\{r_1\}\) is controlled by \(M'(r_0)\) and \(r_1\) is a Lebesgue point of the slicing operator \(\langle {\mathbf {v}}_{m}(R), \rho , \cdot \rangle \) and \(R \cap \rho ^{-1}\{r_1\}\) is \(({\mathscr {H}}^{m-1},m-1)\) rectifiable.
To prove 9.2(a) we cover the slice \(R \cap \rho ^{-1}\{r_1\}\) with cubes of equal size \(\varepsilon > 0\) and apply the deformation Theorem 7.13 again to obtain a map \(g_2 : I \times {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\). We choose \(\varepsilon \) so big that the whole slice \(R \cap \rho ^{-1}\{r_1\}\) does not fill, after the deformation, a single m-dimensional cube, which amounts to setting \(\varepsilon \approx M'(r_0)^{1/(m-1)}\). This ensures that \(g_2(1,\cdot ) [R \cap \rho ^{-1}\{r_1\} ]\) lies in some \(m-2\) dimensional cubical complex. Since \(r_1\) is a Lebesgue point of \(\langle {\mathbf {v}}_{m}(R), \rho , \cdot \rangle \) we may perform a blow-up of the slice using 8.9. Then we make use of the smoothness of \(g_2\) to argue that the push-forward \(g_{2\#}\) is continuous on the space of varifolds so that we can compose \(g_2\) with the blow-up map \(K_{\delta }\) and pass to the limit. Next, we estimate the \({\mathscr {H}}^m\) measure of the blow-up limit only by one term, namely the \({\mathscr {H}}^{m}\) measure of the image of the whole deformation of the slice, i.e., \({\mathscr {H}}^m(g_2[I \times R \cap \rho ^{-1}\{r_1\} ])\). The other terms drop out because \(g_2\) deformed our slice into an \(m-2\) dimensional set. Since \(R \cap \rho ^{-1}\{r_1\}\) is \(({\mathscr {H}}^{m-1},m-1)\) rectifiable we can use the second part of 7.13(g) to estimate \({\mathscr {H}}^m(g_2[I \times R \cap \rho ^{-1}\{r_1\} ])\) by \(\varepsilon M'(r_0) \approx M'(r_0)^{m/(m-1)}\). Finally, we make use of continuity of the mass to choose one deformation from the blow-up sequence (without passing to the limit) for which the desired estimate still holds.
To prove 9.2(a) we proceed similarly but this time we choose \(\varepsilon \approx \iota > 0\) arbitrarily, we deform the part of R lying in the ball \({\mathbf {U}}(a,r_1)\) rather then just the slice \(R \cap \rho ^{-1}\{r_1\}\), and we get two terms in the final estimate. The first term corresponds to \(\iota M'(r_0)\) analogously as before and the second one can be estimated brutally by the \({\mathscr {H}}^m\) measure of the sum of all m dimensional cubes from \({\mathbf {K}}_m\) touching the ball \({\mathbf {B}}(a,r_0 + d + 6 \iota \sqrt{n})\). Later, we use scaling to show that this second term depends only on \(\iota \) and, in 9.3, we choose a specific \(\iota \) depending only on n, m, and F.
We begin with a technical lemma. The estimates in 9.1(a)(b) contain an additional term which includes the parameter d and which shall be later absorbed by other terms. The parameter b below shall be set to \(M'(r_0)\) in most cases.
Lemma 9.1
Let \(S \subseteq {\mathbf {R}}^n\), and \(a \in {\mathbf {R}}^n\), and \(b,d \in (0,\infty )\), and \(r_0 \in (0,\infty )\). Set \(\rho (x) = |x-a|\). Assume \({\mathscr {H}}^m({\mathrm {Clos}}S \cap {\mathbf {B}}(a,r_0 + 4d)) < \infty \) and
-
(a)
There exists a deformation \(D \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n,{\mathbf {R}}^n)\) such that
$$\begin{aligned}&D \in {\mathfrak {D}}({ {\mathbf {U}}(a,r_0 + 4d + 40 \sqrt{n} (\Gamma _{{7.13}}^2 b)^{1/(m-1)}) }), \nonumber \\&\quad {\mathscr {H}}^m( D [S \cap {\mathbf {U}}(a,r_0 + 4d) ]) \le 9 \Gamma _{{7.13}} b d + 50 \Gamma _{{7.13}}^{2m/(m-1)} b^{m/(m-1)} . \end{aligned}$$(62) -
(b)
Suppose \(\iota \in (0,\infty )\) and \(N \in {\mathbf {Z}}\) satisfy \(2^{-N-1} < \iota \le 2^{-N}\). There exists a deformation \(F \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n,{\mathbf {R}}^n)\) such that setting
$$\begin{aligned} \Delta (\rho ,\iota ,r_0,d) = 2^{-Nm} {\mathscr {H}}^0\bigl ( \bigl \{ K \in {\mathbf {K}}_{m}(N) : K \cap {\mathbf {U}}(a,r_0 + d + 6 \iota \sqrt{n}) \ne \varnothing \bigr \}\bigr ) \end{aligned}$$(63)there holds
$$\begin{aligned}&F \in {\mathfrak {D}}({ {\mathbf {U}}(a,r_0 + 4d + 8 \iota \sqrt{n}) }) \nonumber \\&\quad {\mathscr {H}}^m( F [S \cap {\mathbf {U}}(a,r_0 + 4d) ]) \le 9 \Gamma _{{7.13}} b d + 10 \Gamma _{{7.13}}^{2m/(m-1)} \iota b \nonumber \\&\quad +\, \Delta (\rho ,\iota ,r_0,d). \end{aligned}$$(64)
Proof
For brevity define \(A(d) = {\mathbf {U}}(a,r_0 + d) {{\mathrm{\sim }}}{\mathbf {U}}(a,r_0-d)\) for \(d \in (0,\infty )\). Set \(\varepsilon _1 = (5n)^{-1/2} d\) and find \(N_1 \in {\mathbf {Z}}\) such that \(2^{-N_1 - 1} < \varepsilon _1 \le 2^{-N_1}\). Define
Note that \({\mathcal {A}}_1\) is finite. Apply 7.13 with 2, m, m S, \(S \cap A(2d)\), \(2^{-N_1-4}\), \({\mathbf {K}}_n(N_1)\), \({\mathcal {A}}_1\) in place of l, \(m_1\), \(m_2\), \(\Sigma _1\), \(\Sigma _2\), \(\varepsilon \), \({\mathcal {F}}\), \({\mathcal {A}}\) to obtain the map \(g_1 : I \times {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) called “f” there. Observe that
In particular, it follows from 7.13(a) that
Define \(R = g_1(1,\cdot ) [S ]\) and note that \(R \cap {{\mathrm{Int}}}(\mathop {{\textstyle \bigcup }}{\mathcal {A}}_1)\) is a finite sum of m dimensional cubes; in particular it is \(({\mathscr {H}}^m,m)\) rectifiable. Observe also that if \(x \in R \cap A(d)\), then there exists an n dimensional cube \(K \in {\mathcal {A}}_1\) such that \(x \in K\) and there exists \(y \in S\) such that \(g_1(y) = x\) and \(y \in K\) due to 7.13(d). Hence, \(|g(y) - y| \le 2 \varepsilon _1 \sqrt{n} < d\) and, since \({{\mathrm{Lip}}}\rho \le 1\), we get \(y \in A(2d)\). Therefore,
by 7.13(g) and (9.1). Since \(R \cap A(d) \subseteq {{\mathrm{Int}}}(\bigcup {\mathcal {A}}_1)\) is \(({\mathscr {H}}^m,m)\) rectifiable we may employ the co-area formula [19, 3.2.22] together with \({{\mathrm{Lip}}}\rho \le 1\) to obtain
Thus, the Chebyshev inequality gives
Now we see that there exists \(r_1 \in (r_0 - d, r_0 + d)\) such that
and \(r_1\) is a Lebesgue point of the slicing operator \(\langle {\mathbf {v}}_{m}(R), \rho , \cdot \rangle \) (see 8.4) and \(\langle {\mathbf {v}}_{m}(R), \rho , r_1 \rangle \in \mathbf {RV}_{m-1}({\mathbf {R}}^n)\) (see 8.6). For \(\delta \in (0,r_1-r_0+d)\) let \(K_{\delta } = K_{\rho ,r_1,\delta } : {\mathbf {R}}^n \rightarrow I \times {\mathbf {R}}^n\) be defined as in 8.8. Since \(R \cap A(d) \subseteq {{\mathrm{Int}}}(\mathop {{\textstyle \bigcup }}{\mathcal {A}}_1)\) is a finite sum of m dimensional cubes we can apply 8.9 to see that
where \(i_{0}\) and \(i_{1}\) are defined as in 8.8.
Proof of (a): For brevity, if \(K \in {\mathbf {K}}\), let us define \({\widehat{K}}\) to be the n dimensional cube with the same centre as K and side length three times as long as K. Choose \(\varepsilon _2 \in (0,\infty )\) and \(N_2 \in {\mathbf {Z}}\) so that
Define
Apply 7.13 with 1, \(m-1\), B, \(2^{-N_2-4}\), \({\mathbf {K}}_n(N_2)\), \({\mathcal {A}}_2\) in place of l, \(m_1\), \(\Sigma _1\), \(\varepsilon \), \({\mathcal {F}}\), \({\mathcal {A}}\) to obtain the map \(g_2 : I \times {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) called “f” there. We define \(h : I \times {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) by setting
where \(s_{1/100}\) is the function defined in 8.8. Due to our choice of \(\varepsilon _2\) we know, from 7.13(g), that \({\mathscr {H}}^{m-1}(g_2[B ]) < {\mathscr {H}}^{m-1}(K)\) for any \((m-1)\) dimensional cube \(K \in {\mathbf {K}}_{m-1}(N_2)\). We see also that \(g_2[B ]\subseteq {{\mathrm{Int}}}(\mathop {{\textstyle \bigcup }}{\mathcal {A}}_2)\) because \(g_2[B ]\subseteq \mathop {{\textstyle \bigcup }}\{ K \in {\mathbf {K}}_n(N_2) : K \cap B \ne \varnothing \}\) by 7.13(d). Hence, by 7.13(f),
and we obtain
Since h is of class \({\mathscr {C}}^{\infty }\) the push-forward \(h_{\#}\) is continuous on \({\mathbf {V}}_{m}({\mathbf {R}}\times {\mathbf {R}}^n)\) so using (68) together with (70), (71), (72)
Since \(\rho \) is proper we can find a continuous function \(\gamma : {\mathbf {R}}\times {\mathbf {R}}^n \rightarrow {\mathbf {R}}\) with compact support such that \(I \times {\mathbf {U}}(a,r_0 + 4d) \subseteq {{\mathrm{Int}}}\gamma ^{-1}\{1\}\) and use it as a test function for the weak convergence. Thus, recalling that \(\langle {\mathbf {v}}_{m}(R), \rho , r_1 \rangle \in \mathbf {RV}_{m-1}({\mathbf {R}}^n)\) and employing 7.13(g), (67), (66) we obtain
For \(\delta \in (0,r_1 - r_0 + d)\) define \(D_{\delta } = h \circ K_{\delta } \circ g_1(1,\cdot )\). Using (65), (69), (73) we can find \(\delta _0 \in (0,r_1 - r_0 + d)\) such that for all \(\delta \in (0,\delta _0]\)
Observe that
hence, \(D_{\delta } \in {\mathfrak {D}}({{{\mathrm{conv}}}A(4d + 40 \sqrt{n} (\Gamma _{{7.13}}^2 b)^{1/(m-1)})})\) for \(\delta \in (0,\delta _0)\) so setting \(D = D_{\delta _0}\) finishes the proof of (a).
Proof of (b): Recall that if \(K \in {\mathbf {K}}\), then \({\widehat{K}}\) denotes the n dimensional cube with the same centre as K and side length three times as long as K. Let \(\iota \in (0,\infty )\) and \(N \in {\mathbf {Z}}\) satisfy \(2^{-N-1} < \iota \le 2^{-N}\). Set
Apply 7.13 with 2, m, \(m-1\), C, B, \(2^{-N-4}\), \({\mathbf {K}}_n(N)\), \({\mathcal {A}}_3\) in place of l, \(m_1\), \(m_2\), \(\Sigma _1\), \(\Sigma _2\), \(\varepsilon \), \({\mathcal {F}}\), \({\mathcal {A}}\) to obtain the map \(g_3 : I \times {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) called “g” there. Since \(C \subseteq \mathop {{\textstyle \bigcup }}\bigl \{ K \in {\mathbf {K}}_n(N) : K \cap {\mathbf {U}}(a,r_1) \ne \varnothing \bigr \}\) we see that \(g_3(1,\cdot ) [C ]\subseteq {{\mathrm{Int}}}(\mathop {{\textstyle \bigcup }}{\mathcal {A}}_3)\), by 7.13(d), and conclude from 7.13(e) that
Therefore, using (68) and 7.13(g), (66), (67), (74) we get
Hence, there exists \(\delta _0 \in (0,r_1 - r_0 + d)\) such that \(F = g_3 \circ K_{\delta _0} \circ g_1(1,\cdot )\) satisfies the estimate claimed in (b). Moreover, we see that
hence, \(F \in {\mathfrak {D}}({ {\mathbf {U}}(a,r_0 + 4d + 8 \iota \sqrt{n}) })\). \(\square \)
Now, we can prove the pivotal differential inequalities (78) and (80). There is one technical difficulty that needs to be taken care of. To be able to employ minimality of V we need our deformations to be admissible in an open set \(U \subseteq {\mathbf {R}}^n\). In particular, in the proof of (78) we need to perform a deformation onto cubes of side length roughly \(M'(r_0)^{1/(m-1)}\) which might be arbitrarily big. This turns out not to be a problem since big values of the derivative \(M'\) cannot spoil the lower bound on M. More precisely, we will later use the upper bound on M proven in 9.3(a) to overcome this difficulty. For the time being we just assume in 9.2(a) that some upper bound on M holds.
Lemma 9.2
Assume
-
(a)
If \(M(r_0) \le \gamma r_0^m\) or \(M'(r_0) \le (\gamma /\Gamma )^{1-1/m} r_0^{m-1}\) for some \(\gamma \in (0,\infty )\),
$$\begin{aligned} \kappa = \kappa (n,m,\Gamma ,\gamma ) = 1 + (\gamma /\Gamma )^{1/m} \bigl ( 4 \Gamma _{{7.13}}^{(m+1)/(m-1)} + 40 \sqrt{n} \Gamma _{{7.13}}^{2/(m-1)} \bigr ), \end{aligned}$$(77)and \({\mathbf {B}}(a,\kappa r_0) \subseteq U\), then
$$\begin{aligned} M(r_0) \le \Gamma M'(r_0)^{m/(m-1)}. \end{aligned}$$(78) -
(b)
There exists \(\gamma = \gamma (n,m,\iota ,\alpha ,\beta ) \in (1,\infty )\) such that setting
$$\begin{aligned} \kappa = \kappa (n,m,\iota ) = 1 + \iota \bigl (4 \Gamma _{{7.13}}^{(m+1)/(m-1)} + 8 \sqrt{n} \bigr ) \end{aligned}$$(79)if \({\mathbf {B}}(a,\kappa r_0) \subseteq U\), then
$$\begin{aligned} \frac{M(r_0)}{r_0^m} \le \gamma + \Gamma \iota \frac{M'(r_0)}{r_0^{m-1}}. \end{aligned}$$(80)
Proof
Define \(A(d) = \{ x \in {\mathbf {R}}^n : r_0 - d \le \rho (x) < r_0 + d\}\) for \(d \in (0,\infty )\). Choose \(b,d \in (0,\infty )\) so that
It follows from (81) that \(\lim _{i \rightarrow \infty } \Vert {\mathbf {v}}_{m}(S_i)\Vert (A(4d)) = \Vert V\Vert (A(4d))\); hence, recalling (75), we see that there exists \(S \in \{ S_i : i \in {\mathscr {P}}\}\) such that
Proof of (a): If \(M(r_0) \le \gamma r_0^m\) and \(M'(r_0) > (\gamma /\Gamma )^{1-1/m} r_0^{m-1}\), then (78) follows trivially. Thus, we may and shall assume that \(M'(r_0) < (\gamma /\Gamma )^{1-1/m} r_0^{m-1}\). Suppose also \(b \le (\gamma /\Gamma )^{1-1/m} r_0^{m-1}\) and define \(\kappa \) by (77).
Now, recalling (62) and (82), we apply 9.1(a) together with (82) to obtain the deformation \(D \in {\mathfrak {D}}({{\mathbf {U}}(a,\kappa r_0)})\) such that
Since \(D \in {\mathfrak {D}}({U})\) we have \(\Phi _{F}(V) \le \Phi _{F}(D [S ]\cap U)\). Using (84) and (83), and noting that \(D(x) = x\) whenever \(\rho (x) \ge r_0 + 4d\) we see that
Recalling (76), the definition of \(\Gamma \), we see that
Clearly M is non-decreasing so \(M'(r_0) \ge 0\). If \(M'(r_0) > 0\), then the proof of (a) is finished by setting \(b = M'(r_0)\). If \(M'(r_0) = 0\), then we may choose \(b > 0\) arbitrarily small to obtain \(M(r_0) = 0 \le \Gamma M'(r_0) = 0\).
Proof of (b): From (a) we already know that if \(M'(r_0) = 0\), then \(M(r) = 0\) so we may assume \(M'(r_0) > 0\) and set \(b = M'(r_0)\). Define
Apply 9.1(b) with \({\overline{S}}\), \(\overline{d}\), \({\overline{\rho }}\), \({\overline{b}}\), \(\iota \) in place of S, d, \(\rho \), b, \(\iota \) to obtain the deformation \(F \in {\mathfrak {D}}({{\mathbf {U}}(0,\kappa )})\), where \(\kappa = \kappa (n,m,\iota )\) is defined by (79). Combining (82) and (64) we see that
where \(\Delta (\iota ) = \Delta ({\overline{\rho }},\iota ,1,{\overline{d}})\) is defined by (63). Set \(L = \varvec{\tau }_{a} \circ \varvec{\mu }_{r_0} \circ F \circ \varvec{\mu }_{1/r_0} \circ \varvec{\tau }_{-a}\). Since \({\mathbf {B}}(a,\kappa r_0) \subseteq U\) we have \(L \in {\mathfrak {D}}({U})\) so \(\Phi _{F}(V) \le \Phi _{F}(L[S ])\) and we can compute as in (85)
Hence, we may set \(\gamma = 4 \beta / \alpha \Delta (\iota )\). \(\square \)
Theorem 9.3
Assume
Then \(\Delta , \iota \in (0,\infty )\) and the following statements hold.
-
(a)
There exists \(\Gamma = \Gamma (n,m,F,V,U) \in (0,\infty )\) such that for all \(r \in (0,r_0)\)
$$\begin{aligned} r^{-m} {\Vert V\Vert }\,{{\mathbf {U}}(a,r)} \le \max \bigl \{ \Gamma , r_0^{-m} {\Vert V\Vert }\,{{\mathbf {U}}(a,r_0)} \bigr \}. \end{aligned}$$ -
(b)
Define \(\lambda = \lambda (n,m,F,V,U,a) = \max \{ \kappa , \kappa _{9.2(a)}(n,m,\Gamma _{{9.2}}(n,m,F,U,a),\gamma ) \}\), where
$$\begin{aligned} \gamma = \gamma (n,m,F,V,U,a) = \max \bigl \{ \Gamma _{9.3(a)}(n,m,F,V,U) ,\, r_0^{-m} {\Vert V\Vert }\,{{\mathbf {U}}(a,r_0)} \bigr \}. \end{aligned}$$For all \(r \in (0,{{\mathrm{dist}}}(a,{\mathbf {R}}^n {{\mathrm{\sim }}}U) / \lambda )\) we have
$$\begin{aligned} r^{-m} {\Vert V\Vert }\,{{\mathbf {U}}(a,r)} \ge m^{-m} \Gamma _{{9.2}}(n,m,F,U,a)^{1-m}. \end{aligned}$$
Proof
Since F is bounded, it attains its supremum and infimum. Thus, recalling the definition of \(\Gamma _{{9.2}}(n,m,F,U,\cdot )\) we see that \(0< \Delta < \infty \).
Proof of (a): Let \(a \in {{\mathrm{spt}}}\Vert V\Vert \) and \(r \in (0,r_0)\), where \(r_0 = {{\mathrm{dist}}}(a,{\mathbf {R}}^n {{\mathrm{\sim }}}U)/\kappa \). Set \(M(s) = {\Vert V\Vert }\,{{\mathbf {U}}(a,s)}\) for \(s \in (0,\infty )\). Define \(\gamma = \gamma _{9.2}(n, m, \iota , \inf {{\mathrm{im}}}F, \sup {{\mathrm{im}}}F)\) and \(\Gamma = 2\gamma /\varvec{\alpha }(m)\). For each \(s \in (0,r_0)\) for which \(M'(s)\) exists and is finite we may apply 9.2(b) to see that
Now we proceed as in [21, 8.2]. Choose \(\eta > \Gamma \) and assume there exists \(r_1 \in (0,r_0)\) satisfying \(M(r_1) > \eta \varvec{\alpha }(m) r_1^m\). Let \(r_2 \in [r_1,r_0]\) be the largest number in \([r_1,r_0]\) such that \(M(s) \ge \eta \varvec{\alpha }(m) s^m\) for \(s \in [r_1,r_2]\). Since M is non-decreasing we see immediately that \(r_2 > r_1\). Using (86) and the definitions of \(\iota \) and \(\Gamma \), we obtain for \({\mathscr {L}}^1\) almost all \(s \in [r_1,r_2]\)
Hence, \(m M(s) < s M'(s)\) for \({\mathscr {L}}^1\) almost all \(s \in [r_1,r_2]\) which implies that
Using [19, 2.9.19] for each \(s_1,s_2 \in [r_1,r_2]\) with \(s_1 < s_2\) we obtain
which shows that \(s^{-m} M(s)\) is increasing for \(s \in [r_1,r_2]\); thus, \(r_2 = r_0\). Since \(\eta > \Gamma \) could be arbitrary the claim is proven.
Proof of (b): For each \(s \in (0,{{\mathrm{dist}}}(a,{\mathbf {R}}^n {{\mathrm{\sim }}}U)/\lambda )\) for which \(M'(s)\) exists and is finite we may apply 9.2(a) to see that
Employing [19, 2.9.19] we find out that \(M'(s)\) exists and is finite for \({\mathscr {L}}^1\) almost all \(s \in (0,r_0)\) and that
Corollary 9.4
Let F, V, and U be as in 9.3 and \(\delta > 0\). There exist \(\Gamma = \Gamma (n,m,F,V,U,\delta ) > 1\) and \(\kappa = \kappa (n,m,F,V,U,\delta ) > 1\) such that for all \(x \in {{\mathrm{spt}}}\Vert V\Vert \subseteq U\) and \(r \in (0,\infty )\) satisfying \(r < {{\mathrm{dist}}}(x,{\mathbf {R}}^n {{\mathrm{\sim }}}U)/\kappa \) and \({{\mathrm{dist}}}(x,{\mathbf {R}}^n {{\mathrm{\sim }}}U) > \delta \) there holds
In particular, for all \(x \in {{\mathrm{spt}}}\Vert V\Vert \cap E\) we have
Using [19, 2.10.19(1)(3), 2.1.3(5)] and Borel regularity of the Hausdorff measure [19, 2.10.2(1)] we further deduce that there exists \(C = C(n,m,F,V,U,\delta ) > 1\) such that for any Borel set \(A \subseteq \{ x \in U : {{\mathrm{dist}}}(x, {\mathbf {R}}^n {{\mathrm{\sim }}}U) > \delta \}\) we have
10 Rectifiability of the support of the limit varifold
In 10.1 we prove that the support of a \(\Phi _F\) minimising varifold V must be \(({\mathscr {H}}^m,m)\) rectifiable inside any compact set \(K \subseteq U\). Using the density ratio bounds 9.4 we also conclude, in 10.3, that the approximate tangent cones of \(\Vert V\Vert \) coincide with the classical tangent cones of \({{\mathrm{spt}}}\Vert V\Vert \) for all points \(x \in {{\mathrm{spt}}}\Vert V\Vert \subseteq U\). In consequence, the cones \({{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x)\) are in fact m-planes for \({\mathscr {H}}^m\) almost all \(x \in {{\mathrm{spt}}}\Vert V\Vert \).
In the proof of 10.1 we follow the guidelines presented in [3, 2.9(b4), p. 341]. We use only boundedness of F and make no use of ellipticity of F. The proof is done by contradiction. We assume that \({{\mathrm{spt}}}\Vert V\Vert \) is not countably \(({\mathscr {H}}^m,m)\) rectifiable and we look at a density point \(x_0 \in U\) of the unrectifiable part of \({{\mathrm{spt}}}\Vert V\Vert \). We choose a scale \(\rho _1 > 0\) so that the \({\mathscr {H}}^m\) measure of the rectifiable part of \({{\mathrm{spt}}}\Vert V\Vert \cap {\mathbf {B}}(x_0,\rho _1)\) is negligible in comparison to the \({\mathscr {H}}^m\) measure of the unrectifiable part. Then we use the deformation Theorem 7.13 to produce a smooth map \(\phi : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) which deforms \({{\mathrm{spt}}}\Vert V\Vert \cap {\mathbf {B}}(x_0,\rho _1)\) onto an m-dimensional skeleton of some cubical complex. Next, we apply a perturbation argument 4.3 to obtain a map g which almost kills the \({\mathscr {H}}^m\) measure of the unrectifiable part of \({{\mathrm{spt}}}\Vert V\Vert \) keeping the \({\mathscr {H}}^m\) measure of the rectifiable part negligible.
At this point we know that \({\mathscr {H}}^m(g [{{\mathrm{spt}}}\Vert V\Vert \cap {\mathbf {B}}(x_0,\rho _1) ])\) is significantly smaller than \({\mathscr {H}}^m({{\mathrm{spt}}}\Vert V\Vert \cap {\mathbf {B}}(x_0,\rho _1))\) and we want to contradict minimality of V but we do not know whether \({\mathbf {v}}_{m}({{\mathrm{spt}}}\Vert V\Vert ) = V\), i.e., whether \(\Phi _F(V) = \Phi _F({{\mathrm{spt}}}\Vert V\Vert )\). Thus, we look at \(g [S_i ]\), where \(S_i \in {\mathcal {C}}\) is an appropriate minimising sequence (we need to assume it converges in the Hausdorff metric to some compact set \(S \subseteq {\mathbf {R}}^n\) such that \({\mathscr {H}}^m(S \cap U {{\mathrm{\sim }}}{{\mathrm{spt}}}\Vert V\Vert ) = 0\); see 11.2). To compare the measures of \(g [{{\mathrm{spt}}}\Vert V\Vert \cap {\mathbf {B}}(x_0,\rho _1) ]\) and \(g[S_i \cap {\mathbf {B}}(x_0,\rho _1) ]\) we make use of a simple observation: these two sets both lie in the m-dimensional skeleton of a fixed cubical complex and are close in Hausdorff metric so their \({\mathscr {H}}^m\) measures must also be close; see (102).
Theorem 10.1
Assume
Then \({\mathscr {H}}^m({{\mathrm{spt}}}\Vert V\Vert \cap K) < \infty \) for any compact set \(K \subseteq U\) and \({{\mathrm{spt}}}\Vert V\Vert \) is a countably \(({\mathscr {H}}^m,m)\) rectifiable subset of U.
Proof
For \(\delta > 0\) set \(U_{\delta } = \{ x \in U : {{\mathrm{dist}}}(x,{\mathbf {R}}^n {{\mathrm{\sim }}}U) > \delta \}\). Note that for each \(\delta > 0\), by 9.4, there exists some number \(C(\delta ) > 1\) such that
This proves the first part of 10.1. We shall prove the second part by contradiction.
Assume that \({{\mathrm{spt}}}\Vert V\Vert \) is not countably \(({\mathscr {H}}^m,m)\) rectifiable. Then there exists \(\delta > 0\) such that \(E = {{\mathrm{spt}}}\Vert V\Vert \cap U_{\delta }\) is not countably \(({\mathscr {H}}^m,m)\) rectifiable. We decompose E into a disjoint sum \(E = E_r \cup E_u\), where \(E_r\) is \(({\mathscr {H}}^m,m)\) rectifiable, and \(E_u\) is purely \(({\mathscr {H}}^m,m)\) unrectifiable, and \({\mathscr {H}}^m(E_u) > 0\). Employing [19, 2.10.19(2)(4)] we choose \(x_0 \in E_u\) such that
Since F is bounded, there exist \(0< C_1< C_2 < \infty \) such that
From 9.4 we see that there exist numbers \(0< C_3< C_4 < \infty \) (depending on \(\delta \)) such that
Next, we fix a small number \(\varepsilon >0\) such that
Since \({\mathscr {H}}^m(E) < \infty \), we see that 
is a Radon measure; hence,
Employing (87) and (91) we choose \(0< \rho _1< \rho _3 < \rho _2\) such that
Choose \(k \in {\mathscr {P}}\) such that \(2^{-k} \le (\rho _2 - \rho _1) / 16 < 2^{-k+1}\). Define
Then
Next, apply 7.13 with \({\mathbf {K}}_n(k)\), \({\mathcal {A}}\), \(\Sigma _1\), \(\Sigma _2\), \(\Sigma _3\), 3, m, m, m, \(2^{-k+8}\) in place of \({\mathcal {F}}\), \({\mathcal {A}}\), \(\Sigma _1\), \(\Sigma _2\), \(\Sigma _3\), l, \(m_1\), \(m_2\), \(m_3\), \(\varepsilon \) to obtain the map \(f : {\mathbf {R}}\times {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) of class \({\mathscr {C}}^{\infty }\) called “g” there. Define
Recalling 7.13(a, e, g) we see that there exists an open set \(W \subseteq {\mathbf {R}}^n\) such that
For each \(\iota \in (0,1)\) recall 7.14 and apply 4.3 with \({\mathbf {U}}(x_0,\rho _3)\), \(E_u \cap {\mathbf {B}}(x_0,\rho _1)\), \(\phi \), \(\iota \) in place of U, K, f, \(\varepsilon \) to get a diffeomorphism \(\varphi _{\iota }\) of \({\mathbf {R}}^n\). Since \(\phi \) is of class \({\mathscr {C}}^{\infty }\), recalling 4.5, we can find \(\iota > 0\) such that, setting
we obtain
Thus, employing (100), (99), (94), (95) we get
Observe that \(\varphi [S ]\subseteq W\) by (97), (99). Recalling (98), (96), we see that \(g [S \cap {\mathbf {B}}(x_0,\rho _3) ]\subseteq A_m\) so for each \(\iota > 0\) we can find an open set \(V_{\iota } \subseteq {\mathbf {R}}^n\) such that
For \(\iota > 0\) set \(W_{\iota } = W \cap g^{-1} [V_{\iota } ]\) and note that \(W_{\iota }\) is open and \(S \cap {\mathbf {B}}(x_0,\rho _3) \subseteq W_{\iota }\). Since \({d_{{\mathscr {H}},K}}(S_i \cap U, S \cap U) \rightarrow 0\) as \(i \rightarrow \infty \) for all compact sets \(K \subseteq U\) we see that for \(i \in {\mathscr {P}}\) large enough \(S_i \cap {\mathbf {B}}(x_0,\rho _3) \subseteq W_\iota \); thus,
But \(\iota > 0\) can be chosen arbitrarily small, so
Moreover, recalling that \({\mathbf {v}}_{m}(S_i \cap U) \rightarrow V\) as \(i \rightarrow \infty \) and using (93) together with [1, 2.6(2)(d)], (89), (99), and 7.13(g) we obtain
Combining (103) and (104) we get
Note that \({\mathscr {H}}^m(S \cap {\mathbf {B}}(x_0,\rho _2)) = {\mathscr {H}}^m(E \cap {\mathbf {B}}(x_0,\rho _2))\). In consequence, using (88), (101), (95), (94), (92), and finally (88), (89), (90), we get
where
Recall that \(g(x) = x\) for \(x \in {\mathbf {R}}^n {{\mathrm{\sim }}}{\mathbf {U}}(x_0,\rho _2)\) and \(g [{\mathbf {B}}(x_0,\rho _2) ]\subseteq {\mathbf {B}}(x_0,\rho _2)\) and \({\mathbf {v}}_{m}(S_i \cap U) \rightarrow V\) as \(i \rightarrow \infty \). Hence, using (105), (106), and (93) together with (89), we obtain
Clearly \(g \in {\mathfrak {D}}({U})\) so \(g[S_i ]\in {\mathcal {C}}\) for each \(i \in {\mathscr {P}}\) but (107) yields
which contradicts \(\Phi _F(V) = \inf \{ \Phi _F(R \cap U) : R \in {\mathcal {C}} \}\). \(\square \)
Lemma 10.2
Let \(\mu \) be a Radon measure over \({\mathbf {R}}^n\) and \(a \in {\mathbf {R}}^n\). Assume there exist \(C \in (0,\infty )\) and \(r_0 \in (0,\infty )\) such that for all \(x \in {\mathbf {B}}(a,r_0)\) and \(r \in (0,r_0)\)
Then \({{\mathrm{Tan}}}^m(\mu ,a) = {{\mathrm{Tan}}}({{\mathrm{spt}}}\mu , a)\), i.e, the approximate tangent cone of \(\mu \) at a equals the classical tangent cone of the support of \(\mu \) at a.
Proof
Following [19, 3.2.16] if \(a \in {\mathbf {R}}^n\), and \(\varepsilon \in (0,1)\), and \(v \in {\mathbf {R}}^n\), then we define the cone
Notice that if \(|v| < \varepsilon \), then \({\mathbf {E}}(a,v,\varepsilon ) = {\mathbf {R}}^n\) and if \(0 < \varepsilon \le |v|\), then we may set
Let \(S = {{\mathrm{spt}}}\mu \). By definition (see [19, 3.2.16, 3.1.21]) we have
Clearly \({{\mathrm{Tan}}}^m(\mu ,a) \subseteq {{\mathrm{Tan}}}(S,a)\) so we only need to show the reverse inclusion. Let \(v \in {{\mathrm{Tan}}}(S,a)\) and \(\varepsilon \in (0,1)\) be such that \(\varepsilon \le |v|\). From (109) we see that there exists a sequence \(\{ x_k \in {\mathbf {R}}^n : k \in {\mathscr {P}}\}\) such that
Let us set \(r_k = |x_k - a|\) for \(k \in {\mathscr {P}}\). Observe that whenever \(1/k < \min \{ r_0, \varepsilon \} /2\), and \(r_k < \varepsilon /2\), and \(z \in {\mathbf {B}}(x_k,r_k \varepsilon /2)\), then setting \(s = (x_k - a) \bullet v |x_k - a|^{-2}\) we obtain
Therefore,
Since \(0 < \varepsilon \le |v|\) could be chosen arbitrarily, we see that \(v \in {{\mathrm{Tan}}}^m(\mu ,a)\). \(\square \)
Remark 10.3
Let U and V be as in 10.1 and set \(E = {{\mathrm{spt}}}\Vert V\Vert \subseteq U\). Then for each \(a \in E\) one can find \(0< r_0 < {{\mathrm{dist}}}(a, {\mathbf {R}}^n {{\mathrm{\sim }}}U)\) such that 
satisfies the conditions of 10.2 at a; thus, for all points \(a \in E\) we have \({{\mathrm{Tan}}}^m(\Vert V\Vert ,a) = {{\mathrm{Tan}}}(E,a)\). In particular, \({{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x) \in {\mathbf {G}}(n,m)\) for \({\mathscr {H}}^m\) almost all \(x \in {{\mathrm{spt}}}\Vert V\Vert \subseteq U\); see [19, 3.2.19].
Lemma 10.4
Let \(G \subseteq {\mathbf {R}}^n\) be open and bounded, \(S \subseteq {\mathbf {R}}^n\) be closed with \({\mathscr {H}}^m(S \cap G) < \infty \) and such that \({\mathscr {H}}^m(S \cap {{\mathrm{Bdry}}}G) = 0\), and let \(\varepsilon > 0\). Decompose \(S \cap G\) into a disjoint sum \(S \cap G = S_u \cup S_r\), where \(S_u\) is purely \(({\mathscr {H}}^m,m)\) unrectifiable and \(S_r\) is \(({\mathscr {H}}^m,m)\) rectifiable.
There exists a number \(\Gamma = \Gamma (n,m) \ge 1\) and a \({\mathscr {C}}^{\infty }\) smooth map \(g \in {\mathfrak {D}}({G})\) such that
Proof
Let \({\mathcal {F}} = \mathbf {WF}(G)\) be the Whitney family associated to G. If \({\mathscr {H}}^m(S_u) > 0\), set \(M = {\mathscr {H}}^m(S_u)\) and if \({\mathscr {H}}^m(S_u) = 0\), set \(M = 1\). Choose \(N \in {\mathscr {P}}\) so that
Define
where \({{\widetilde{Q}}} = \mathop {{\textstyle \bigcup }}\{ R \in {\mathcal {F}} : R \cap Q \ne \varnothing \}\) for \(Q \in {\mathcal {F}}\). In particular, we obtain
Apply the deformation Theorem 7.13 with \({\mathcal {F}}\), \({\mathcal {A}}\), \(S_r\), 1, m, \(2^{-N-30}\) in place of \({\mathcal {F}}\), \({\mathcal {A}}\), \(\Sigma _1\), l, m, \(\varepsilon \) to obtain the map \(f \in {\mathfrak {D}}({G})\) of class \({\mathscr {C}}^{\infty }\) called “g” there. Let \(\omega \) be the modulus of continuity of \(\Vert \mathrm {D}f\Vert \) as defined in (20) and find \({\bar{\varepsilon }} > 0\) such that \(\omega ({\bar{\varepsilon }}) \le 1\) and \({\bar{\varepsilon }} < 2^{-100} \varepsilon \). Next, recall 7.14 to apply the perturbation Lemma 4.3 with
and obtain the map \(\rho \) called “\(\rho _{\varepsilon }\)” there. Set \(g = f \circ \rho \) and \(A = \mathop {{\textstyle \bigcup }}{\mathcal {A}} + {\mathbf {B}}(0,2^{-N-30})\) and \(\Gamma = 2^{2m-1} (\Gamma _{{7.13}} + 1)\). To estimate \({\mathscr {H}}^m(g [S_r ])\) we employ 7.12 and 4.5
To estimate \({\mathscr {H}}^m(g [S_u ])\) we employ 4.3 and (110)
11 Unit density of the limit varifold
In this section we finish the proof of 3.20.
We first prove a general “hair-combing” Lemma 11.1, which allows to choose a sequence of sets \(\{ S_i : i \in {\mathscr {P}}\}\) such that \({\mathbf {v}}_{m}(S_i \cap U) \rightarrow V \in {\mathbf {V}}_{m}(U)\) and, additionally, \(S_i\) converge locally in Hausdorff metric to some set S such that \({\mathscr {H}}^m(S \cap U {{\mathrm{\sim }}}{{\mathrm{spt}}}\Vert V\Vert ) = 0\). This is achieved by using first the deformation Theorem 7.13 inside Whitney type cubes covering \(U {{\mathrm{\sim }}}{{\mathrm{spt}}}\Vert V\Vert \) and then applying the Blaschke Selection Theorem. Since the deformed sets lie in a fixed grid of cubes we know that the “hair” does not accumulate in the limit.
After that, we basically follow the guidelines presented in [3, 3.2(d), p. 348 paragraph starting with “We now verify that...”, p. 349 l. 10–12] to prove that \({{\mathrm{VarTan}}}(V,x) = \{ {\mathbf {v}}_{m}({{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x)) \}\) for each \(x \in {{\mathrm{spt}}}\Vert V\Vert \subseteq U\) such that \({{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x) \in {\mathbf {G}}(n,m)\) is an m-plane and \(\varvec{\Theta }^m(\Vert V\Vert ,x)\) exits and is finite. That means, we take a sequence of radii \(r_j\) converging to 0 and look at the blow-up limit \((\varvec{\mu }_{1/r_j})_{\#}V\). At each scale we use a smooth deformation to project the part of V inside \({\mathbf {B}}(x,r_j)\) onto \(x+{{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x)\). Then we use ellipticity of F and minimality of V to compare V with the deformed V and conclude that \(\varvec{\Theta }^m(\Vert V\Vert ,x) = 1\). Of course we cannot actually work with V itself but we need to always look at the minimising sequence, because \({{\mathrm{spt}}}\Vert V\Vert \) might not be a member of the good class \({\mathcal {C}}\). After proving that \(\varvec{\Theta }^m(\Vert V\Vert ,x) = 1\) we still need to show that \(T = {{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x)\) for V almost all \((x,T) \in U \times {\mathbf {G}}(n,m)\). To this end we employ the area formula and a well known relation between the tilt-excess and the measure-excess (see 11.4). Since the measure-excess vanishes in the limit, so does the tilt-excess and the theorem is proven.
Since in the definition of ellipticity 3.16(b) we use more general maps then admissible deformations defined in 3.1 we need to prove that in our case one can replace the former with the latter. We do that in 11.5.
Also because we admit unrectifiable competitors and we used \(\Psi _F\) instead of \(\Phi _F\) in 3.16 but, in the end, we want to minimise \(\Phi _F\) so we need to show that the \({\mathscr {H}}^m\) measure of the unrectifiable part of any minimising sequence vanishes in the limit. This is done in 11.7.
We emphasis that the proof of 11.8(b) is the only place in the whole paper where we make use of ellipticity of F.
Lemma 11.1
Let \(U \subseteq {\mathbf {R}}^n\) be open. Assume \(\{ S_i \subseteq {\mathbf {R}}^n : i \in {\mathscr {P}}\}\) is a sequence of closed sets such that \({\mathscr {H}}^m(S_i \cap U) < \infty \) for \(i \in {\mathscr {P}}\) and there exists a limit \(V = \lim _{i \rightarrow \infty } {\mathbf {v}}_{m}(S_i \cap U) \in {\mathbf {V}}_{m}(U)\).
Then there exist a closed set \(X \subseteq {\mathbf {R}}^n\), and \(g_i \in {\mathfrak {D}}({U})\), and a subsequence \(\{ S_i' : i \in {\mathscr {P}}\}\) of \(\{ S_i : i \in {\mathscr {P}}\}\) such that, setting \(E = {{\mathrm{spt}}}\Vert V\Vert \cup ({\mathbf {R}}^n {{\mathrm{\sim }}}U)\) and \(X_i = g_i[S_i' ]\) for \(i \in {\mathscr {P}}\), we obtain
Furthermore, if E is bounded and \(S_i\) is compact for each \(i \in {\mathscr {P}}\) and \(\sup \{ {\mathscr {H}}^m(S_i \cap U) : i \in {\mathscr {P}}\} < \infty \), then
Proof
Let \({\mathcal {F}} = \{ Q_j : j \in {\mathscr {P}}\} = \mathbf {WF}(U {{\mathrm{\sim }}}{{\mathrm{spt}}}\Vert V\Vert )\) be the Whitney family defined in 7.5. For brevity of the notation, if \(Q \in {\mathcal {F}}\), we define
and we set \({{\widetilde{Q}}} = \mathop {{\textstyle \bigcup }}{\text {N}}(Q,1)\). Since \({\mathbf {v}}_{m}(S_i \cap U) \rightarrow V\) in \({\mathbf {V}}_{m}(U)\) as \(i \rightarrow \infty \), using [1, 2.6.2(c)], we see that
Set \(S_i^0 = S_i\) for \(i \in {\mathscr {P}}\) and define inductively \(S_i^j\) for \(j \in {\mathscr {P}}\) by requiring that \(\{ S_i^j : i \in {\mathscr {P}}\}\) be a subsequence of \(\{S_i^{j-1} : i \in {\mathscr {P}}\}\) satisfying
For \(j \in {\mathscr {P}}\) define \(P_j = S_j^j\) and \({\mathcal {A}}_j \subseteq \mathcal F\) to consist of all the cubes \(Q \in {\mathcal {F}}\) with \(Q \subseteq {\mathbf {B}}(0,2^j)\) and satisfying
Clearly \({\mathcal {A}}_j\) are finite. For each \(j \in {\mathscr {P}}\) apply 7.13 with \({\mathcal {F}}\), \({\mathcal {A}}_j\), \(P_j\), m, 1, \(2^{-j-8}\) in place of \({\mathcal {F}}\), \({\mathcal {A}}\), \(\Sigma _1\), \(m_1\), l, \(\varepsilon \) to obtain the map \({\bar{f}}_j \in {\mathscr {C}}^{\infty }({\mathbf {R}}\times {\mathbf {R}}^n,{\mathbf {R}}^n)\) called “f” there. Set
We shall prove that \(\lim _{j \rightarrow \infty } {\mathbf {v}}_{m}(W_j \cap U) = V\) in \({\mathbf {V}}_{m}(U)\).
Let \(\varphi \in {\mathscr {K}}(U \times {\mathbf {G}}(n,m))\) and for \(j \in {\mathscr {P}}\) let \(\zeta _j \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n,[0,1])\) be such that
This choice ensures
We set \({\bar{\zeta }}_j(x,T) = \zeta _j(x)\) for \((x,T) \in {\mathbf {R}}^n \times {\mathbf {G}}(n,m)\) and then
Since \(\{ P_j : j \in {\mathscr {P}}\}\) is a subsequence of \(\{ S_j : j \in {\mathscr {P}}\}\) and \({\mathbf {v}}_{m}(S_j \cap U) \rightarrow V\) in \({\mathbf {V}}_{m}(U)\) as \(j \rightarrow \infty \), we only need to show that \(\lim _{j \rightarrow \infty } {\mathbf {v}}_{m}(W_j \cap U)(\varphi {\bar{\zeta }}_j) = 0\) and \(\lim _{j \rightarrow \infty } {\mathbf {v}}_{m}(P_j \cap U)(\varphi {\bar{\zeta }}_j) = 0\). Set
then \({\mathcal {G}}\) and \({\mathcal {J}}\) are finite and do not depend on \(j \in {\mathscr {P}}\). Moreover, \({{\mathrm{spt}}}\varphi \cap {{\mathrm{spt}}}\zeta _j \subseteq \mathop {{\textstyle \bigcup }}\{ {{\widetilde{Q}}} : Q \in {\mathcal {A}}_j \cap {\mathcal {G}} \} \cup \mathop {{\textstyle \bigcup }}\{ {{\widetilde{Q}}} : Q \in {\mathcal {J}} \}\), and \({{\mathrm{dist}}}(\mathop {{\textstyle \bigcup }}\{ {{\widetilde{Q}}} : Q \in {\mathcal {J}} \}, {{\mathrm{spt}}}\Vert V\Vert ) > 0\); hence,
To deal with \({\mathbf {v}}_{m}(W_j \cap U)(\varphi {\bar{\zeta }}_j)\) we first note that whenever \(Q \in {\mathcal {F}}\), then, recalling 7.13(g),
where \(\Delta = 4^n \ge \sup \{ {\mathscr {H}}^0({\text {N}}(T,1)) : T \in {\mathcal {F}} \}\). Now, we can estimate as in (112) using (113)
This finishes the proof that \(\lim _{j \rightarrow \infty } {\mathbf {v}}_{m}(W_j \cap U) = V\).
Let \(\{ K_i \subseteq U : i \in {\mathscr {P}}\}\) be a sequence of compacts sets such that \(\mathop {{\textstyle \bigcup }}\{ K_i \subseteq U : i \in {\mathscr {P}}\} = U\) and \(K_i \subseteq K_{i+1}\) for \(j \in {\mathscr {P}}\). We set \(W^0_j = W_j\) for \(j \in {\mathscr {P}}\). For \(i \in {\mathscr {P}}\) we define \(\{ W^i_j : j \in {\mathscr {P}}\}\) to be a subsequence of \(\{ W^{i-1}_j : j \in {\mathscr {P}}\}\) such that the sequence \(\{ W^i_j \cap K_i : j \in {\mathscr {P}}\}\) converges in the Hausdorff metric to some compact set \(F_i\)–this can be done employing the Blaschke Selection Theorem; see [34]. Finally, we set \(R_i = W^i_i\) for \(i \in {\mathscr {P}}\) and \(R = \mathop {{\textstyle \bigcup }}\{ F_i : i \in {\mathscr {P}}\}\). We see that for any compact set \(K \subseteq U\) we have \({d_{{\mathscr {H}},K}}(R_i \cap U, R \cap U) \rightarrow 0\) as \(i \rightarrow \infty \).
Now, we need to show that \({\mathscr {H}}^m(R \cap U {{\mathrm{\sim }}}{{\mathrm{spt}}}\Vert V|) = 0\). It will be enough to prove that \({\mathscr {H}}^m(R \cap Q) = 0\) for each \(Q \in {\mathcal {F}}\). Recall that \(R \cap Q\) is the limit, in the Hausdorff metric, of some subsequence of \(\{ f_j [P_j ]\cap Q : j \in {\mathscr {P}}\}\). We claim that for big enough \(j \in {\mathscr {P}}\) the set \(f_j [P_j ]\cap Q\) lies in the \((m-1)\) dimensional skeleton of \({\mathcal {F}}\), i.e., \(f_j [P_j ]\cap Q \subseteq \mathop {{\textstyle \bigcup }}{\mathbf {K}}_{m-1} \cap \mathbf {CX}({\mathcal {F}})\), which implies that \(R \cap Q \subseteq \mathop {{\textstyle \bigcup }}{\mathbf {K}}_{m-1} \cap \mathbf {CX}({\mathcal {F}})\) and \({\mathscr {H}}^m(R \cap Q) = 0\). To prove our claim let \(j_0, j_1, j_2 \in {\mathbf {Z}}\) be such that \(Q = Q_{j_0}\) and \({\mathbf {l}}(Q) = 2^{-j_1}\) and \(Q \subseteq {\mathbf {B}}(0,2^{j_2})\), and assume \(j > \max \{j_0,j_1,j_2\}\).
In case \({{\widetilde{Q}}} \cap P_j = \varnothing \) we have \(f_j [P_j ]\cap Q = \varnothing \), by 7.13(d), and there is nothing to prove. Thus, we assume that \({{\widetilde{Q}}} \cap P_j \ne \varnothing \) which implies that \(Q \in {\mathcal {A}}_j\) due to (111) and \(j > \max \{ j_0, j_1, j_2 \}\). For \(R \in {\text {N}}(Q,1)\) such that \(R \cap P_j \ne \varnothing \) we estimate using 7.13(g)
If \(R \cap P_j \ne \varnothing \) and \(R \in {\text {N}}(Q,1)\), then it follows that \(R \subseteq {{\mathrm{Int}}}\mathop {{\textstyle \bigcup }}{\mathcal {A}}_j\); hence, combining (114) with 7.13(f), we see that \(f_j [P_j ]\cap R \subseteq \mathop {{\textstyle \bigcup }}{\mathbf {K}}_{m-1} \cap \mathbf {CX}({\mathcal {F}})\). Next, observe that
Let \(K \subseteq {\mathbf {R}}^n\) be compact. Observe that for each \(k \in {\mathscr {P}}\), there are only finitely many cubes in \({\mathcal {F}}\) which touch K and have side length at least \(2^{-k}\). If \(j_0 \in {\mathscr {P}}\) is the maximal index of such cube and \(j_1 \in {\mathscr {P}}\) is such that \(Q_{j_0} \subseteq {\mathbf {B}}(0,2^{j_1})\), then for \(j > \max \{j_0,j_1,k\}\) the estimate (114) holds for any \(R \in {\text {N}}(Q,1)\) whenever \(Q \in {\mathcal {F}}\) satisfies \({\mathbf {l}}(Q) \ge 2^{-k}\) and \(Q \cap K \ne \varnothing \). In consequence, as in (115),
Recalling that \({\mathcal {F}} = \mathbf {WF}(U {{\mathrm{\sim }}}{{\mathrm{spt}}}\Vert V\Vert )\) was the Whitney family associated to \({\mathbf {R}}^n {{\mathrm{\sim }}}E\), we see that there exists \(\{ r_j \in (0,\infty ) : j \in {\mathscr {P}}\}\) such that \(r_j \downarrow 0\) as \(j \rightarrow \infty \) and
Observe that for each \(i \in {\mathscr {P}}\) there exists \(j(i) \in {\mathscr {P}}\) such that \(R_i = f_{j(i)}[P_{j(i)} ]\), so setting \(S'_i = P_{j(i)}\) for \(i \in {\mathscr {P}}\) we obtain a subsequence of \(\{S_i : i \in {\mathscr {P}}\}\). Hence, we may finish the proof of the first part of 11.1 by setting \(X = R\) and \(g_i = f_{j(i)}\) and \(X_i = R_i\) for \(i \in {\mathscr {P}}\).
Assume now that E is bounded and \(S_i\) is compact for \(i \in {\mathscr {P}}\) and \(\sup \{ {\mathscr {H}}^m(S_i \cap U) : i \in {\mathscr {P}}\} < \infty \). In this case we need to further modify the sets \(R_i\) to ensure that all the resulting sets \(X_i\) are bounded. To this end we shall simply project the sets \(R_j\) onto the cube \([-M_1,M_1]^n\). Since our definition of admissible mappings allows only for deformations inside convex sets we need to perform the projection in several steps. Using the fact that outside \([-M_1,M_1]^n\) the sets \(R_j\) lie in the \((m-1)\) dimensional skeleton of \({\mathcal {F}}\) we will deduce that the projected sets \(X_j\) give rise to the same varifolds as \(R_j\), i.e., \({\mathbf {v}}_{m}(R_j \cap U) = {\mathbf {v}}_{m}(X_j \cap U)\).
Suppose there exists \(M_0 > 1\) such that
Then \(\sup \{ {\mathscr {H}}^m(R_i \cap U) : i \in {\mathscr {P}}\} < M_0\) and we can find \(M_1 > 2^{10} M_0\) such that
Let \(e_1,\ldots ,e_n\) be the standard basis of \({\mathbf {R}}^n\). For \(i \in \{ -n, \ldots , -1, 1, \ldots , n\}\) and \(j \in {\mathscr {P}}\) we define
-
\(L_i = \{ x \in {\mathbf {R}}^n : x \bullet e_{|i|} = \tfrac{i}{|i|} M_1 \}\),
-
\(H_i = \{ x \in {\mathbf {R}}^n : x \bullet e_{|i|} \tfrac{i}{|i|} \ge M_1 \}\),
-
\(p_i\) to be the affine orthogonal projection onto the affine plane \(L_i\),
-
\(Y_{j,i} \subseteq H_i\) to be a a large cube containing \([-M_1,M_1]^n \cap L_i\) and such that
$$\begin{aligned} R_j \cap H_i \subseteq Y_{j,i}, \end{aligned}$$ -
\(\varphi _{i,j} \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n)\) to be a cut-off function such that \(0 \le \varphi _{i,j}(x) \le 1\) for \(x \in {\mathbf {R}}^n\) and
$$\begin{aligned} Y_{j,i} \subseteq \varphi _{j,i}^{-1} \{1\} \quad \text {and} \quad {{\mathrm{spt}}}\varphi _{j,i} \subseteq {\mathbf {R}}^n {{\mathrm{\sim }}}[-2^{-1}M_1, 2^{-1}M_1]^n \text { is compact and convex}, \end{aligned}$$ -
\(h_{j,i} = p_i \varphi _{j,i} + (\mathbb {1}_{{\mathbf {R}}^n} - \varphi _{j,i}) \mathrm {id}_{{\mathbf {R}}^n} \in {\mathfrak {D}}({U})\),
-
\(h_j = h_{j,-n} \circ \cdots \circ h_{j,-1} \circ h_{j,1} \circ \cdots \circ h_{j,n} \in {\mathfrak {D}}({U})\).
We set \(X_i = h_i [R_i ]\) and \(g_i = h_i \circ f_{j(i)}\) for \(i \in {\mathscr {P}}\). Clearly \(X_j \subseteq [-M_1,M_1]^n\) for each \(j \in {\mathscr {P}}\) so, employing the Blaschke Selection Theorem, we can assume that \(X_j\) converges in the Hausdorff metric to some compact set X. Observe that if \(i,j \in {\mathscr {P}}\) and \(R_i = f_j[P_j ]\) and \(Q \in {\mathcal {F}}\) is such that \(Q \cap P_j \ne \varnothing \) and \(Q \cap {\mathbf {R}}^n {{\mathrm{\sim }}}[-2^{-1} M_1, 2^{-1} M_1]^n \ne \varnothing \), then \({\mathbf {l}}(Q)^m> \Gamma _{{7.13}} M_0 > 1\) and \(Q \in {\mathcal {A}}_j\) and \(\Gamma _{{7.13}} {\mathscr {H}}^m(P_j \cap \mathop {{\textstyle \bigcup }}{\text {N}}(Q,3)) \le \Gamma _{{7.13}} M_0 < {\mathbf {l}}(Q)^m\); hence, \(f_j [P_j ]\cap Q \subseteq \mathbf {CX}({\mathcal {F}}) \cap {\mathbf {K}}_{m-1}\), by 7.13(f), and \({\mathscr {H}}^m(f_j [P_j ]\cap Q) = 0\) so \({\mathscr {H}}^m(h_i [R_i ]\cap Q) = 0\). Since \(X_i \cap [-2^{-1} M_1, 2^{-1} M_1]^n = R_i \cap [-2^{-1} M_1, 2^{-1} M_1]^n\) for \(i \in {\mathscr {P}}\), we see that \({\mathbf {v}}_{m}(R_i \cap U) = {\mathbf {v}}_{m}(X_i \cap U)\) for \(i \in {\mathscr {P}}\) and \({\mathbf {v}}_{m}(X_i \cap U) \rightarrow V \in {\mathbf {V}}_{m}(U)\) as \(i \rightarrow \infty \). \(\square \)
Corollary 11.2
Assume
Then there exist \(\{ S_i : i \in {\mathscr {P}}\} \subseteq {\mathcal {C}}\), \(S \in {\mathcal {C}}\), \(V \in {\mathbf {V}}_{m}(U)\), and \(E \subseteq {\mathbf {R}}^n\) such that
Furthermore, if \(B = {\mathbf {R}}^n {{\mathrm{\sim }}}U\) is compact and there exists a minimising sequence consisting of compact elements of \({\mathcal {C}}\), then
Proof
Let \(\{ R_i : i \in {\mathscr {P}}\}\) be any minimising sequence in \({\mathcal {C}}\), i.e.,
Observe that since F is bounded and \(\mu \) is finite we have \({\mathscr {H}}^m(R_i \cap U) < 2 (\inf {{\mathrm{im}}}F)^{-1} \mu \) for all but finitely many \(i \in {\mathscr {P}}\)–we shall assume it holds for all \(i \in {\mathscr {P}}\). In consequence, we can choose a subsequence \(\{ R_i' : i \in {\mathscr {P}}\}\) of \(\{ R_i : i \in {\mathscr {P}}\}\) such that \({\mathbf {v}}_{m}(R_i' \cap U)\) converges as \(i \rightarrow \infty \) to some \(V \in {\mathbf {V}}_{m}(U)\). If \(B = {\mathbf {R}}^n {{\mathrm{\sim }}}U\) is compact, then we use 9.4 too see that \({{\mathrm{spt}}}\Vert V\Vert \) must be bounded; hence, E is bounded. Next, we apply 11.1 to \(\{ R_i' : i \in {\mathscr {P}}\}\) to obtain a subsequence \(\{ P_i : i \in {\mathscr {P}}\}\) of \(\{ R_i' : i \in {\mathscr {P}}\}\) and maps \(\{ g_i \in {\mathfrak {D}}({U}) : i \in {\mathscr {P}}\}\) and \(S \in \mathcal C\). Finally, we set \(S_i = g_i[P_i ]\). \(\square \)
Remark 11.3
Let us summarise what we know so far.
Under the assumptions of 11.2 we obtain a minimising sequence \(\{ S_i : i \in {\mathscr {P}}\} \subseteq {\mathcal {C}}\), and \(V \in {\mathbf {V}}_{m}(U)\), and \(S \in {\mathcal {C}}\) satisfying
Next, employing 10.1 we see that \({{\mathrm{spt}}}\Vert V\Vert \) is countably \(({\mathscr {H}}^m,m)\) rectifiable and has locally finite \({\mathscr {H}}^m\) measure. In particular, 
for \({\mathscr {H}}^m\) almost all \(x \in {{\mathrm{spt}}}\Vert V\Vert \subseteq U\) by [19, 3.2.19] so, using 9.4 and [19, 2.8.18, 2.9.5], also the density \(\varvec{\Theta }^m(\Vert V\Vert ,x)\) exists and is finite for \(\Vert V\Vert \) almost all \(x \in U\). Recalling 10.3 we see that for \({\mathscr {H}}^m\) almost all \(x \in {{\mathrm{spt}}}\Vert V\Vert \) the classical tangent cone \({{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x)\) is an m-dimensional subspace of \({\mathbf {R}}^n\).
In the next lemma we relate the \(L^2\) tilt-excess to the measure-excess; see (146). Similar upper bound was also proven in [31, 3.13].
Lemma 11.4
Let \(P,Q \in {\mathbf {G}}(n,m)\). Then
Proof
Employ [1, 8.9(3)] to find \(u \in Q\) such that \(|u| = 1\) and \(\Vert {P}_\natural - {Q}_\natural \Vert = |P_\natural ^\perp u|\). Let \(u_1, \ldots , u_m\) be an orthonormal basis of Q such that \(u_1 = u\). We have
In consequence, since \((1-x)^{1/2} \le 1 - \frac{1}{2} x\) for \(x \in [0,1]\), we obtain
Next, we shall derive the upper estimate. Let \(q \in {\mathbf {O}}^*({n},{m})\) be such that \({{\mathrm{im}}}q^* = Q\). Since \({P}_\natural \circ {Q}_\natural = {Q}_\natural - P_\natural ^\perp \circ {Q}_\natural \) and \(q^* \circ q = {Q}_\natural \) and \((P_\natural ^\perp )^* = P_\natural ^\perp \) we obtain, using [19, 1.7.6. 1.7.9, 1.4.5],
where \(E = \sum _{j=2}^m (-1)^j \bigl | {\textstyle \bigwedge }_j \bigl ( P_\natural ^\perp \circ q^* \bigr ) \bigr |^2\). Note that \(1 - x \le (1-x)^{1/2}\) for \(x \in [0,1]\); hence,
Employing [19, 1.7.6, 1.7.9, 1.3.2] together with [1, 8.9(3)] we get
If \(\Vert {P}_\natural - {Q}_\natural \Vert ^2 \le 2^{-(m+2)}\), then
If \(2^{-(m+2)} < \Vert {P}_\natural - {Q}_\natural \Vert ^2 = \Vert P_\natural ^\perp \circ {Q}_\natural \Vert ^2 \le 1\), then
Combining (116), (117), (118), and (119) we obtain
The following lemma relates Lipschitz maps from the definition of an elliptic integrand 3.16 to admissible maps defined in 3.1. Given S and D as in 3.16 and a Lipschitz deformation which maps S onto the relative boundary of D the lemma provides conditions on the set S under which one can construct an admissible map deforming S onto the relative boundary of D.
Lemma 11.5
Assume
Suppose there exists a map \(f : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) satisfying
Then there exist \(\Gamma = \Gamma ({{\mathrm{Lip}}}f,\delta ) \in (0,\infty )\) and a map \(g : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) such that
Moreover for each \(\varepsilon > 0\) there exists \(h \in {\mathfrak {D}}({{\mathbf {U}}(0,1+\varepsilon )})\) such that
Proof
Define a map \({\bar{g}} : {\mathbf {R}}^n {{\mathrm{\sim }}}{\mathbf {U}}(0,1) \cup S \rightarrow {\mathbf {R}}^n\) by setting
We shall check that \({{\mathrm{Lip}}}{\bar{g}} < \infty \). Since \(S {{\mathrm{\sim }}}(R + {\mathbf {B}}(0,\delta ))\) does not touch \({{\mathrm{Bdry}}}{\mathbf {B}}(0,1)\) we see that \({\bar{g}}\) is Lipschitz continuous on each of the sets
Note, however, that the Lipschitz constant of \({\bar{g}}\) on the last set depends on \(\delta \). Now, it suffices to estimate \(|{\bar{g}}(x) - {\bar{g}}(y)|\) for \(x \in S \cap (R + {\mathbf {B}}(0,\delta )) \subseteq T\) and \(y \in {\mathbf {R}}^n {{\mathrm{\sim }}}{\mathbf {U}}(0,1)\). Note that \(x/|x| \in R\), so \(f(x/|x|) = x/|x|\) and
Clearly \(|x - x/|x|| \le 2 |x-y|\), so \({{\mathrm{Lip}}}{\bar{g}} < \infty \).
Next, we extend \({\bar{g}}\) to a Lipschitz map \({{\tilde{g}}} : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) using a standard procedure; see [15, 3.1.1]. Let \(L = {{\mathrm{Lip}}}{{\tilde{g}}} \in [1,\infty )\). Since \({{\tilde{g}}}(x) = x\) for \(x \in {{\mathrm{Bdry}}}{\mathbf {B}}(0,1)\) we know \({{\tilde{g}}} [{\mathbf {B}}(0,1) ]\subseteq {\mathbf {B}}(0,2L + 1)\). Define the map \(l : {\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) by setting
where \(\sigma (t) = (28 L t - 4)/3\). Finally set
To prove the second part of the lemma assume \(\varepsilon \in (0,2^{-7})\). Choose \(\iota \in (0,2^{-12}\varepsilon )\) so small that \(g[{\mathbf {B}}(x,\iota )]\subseteq {\mathbf {U}}(g(x),2^{-12}\varepsilon )\) for all \(x \in S\). Let \(\varphi : {\mathbf {R}}^n \rightarrow {\mathbf {R}}\) be a mollifier such that \({{\mathrm{spt}}}\varphi \in {\mathbf {U}}(0,\iota )\) and \(\varphi (x) = {\bar{\varphi }}(|x|)\) for some smooth function \({\bar{\varphi }} : {\mathbf {R}}\rightarrow [0,\infty )\). Let \({\bar{h}} = \varphi * g\). Clearly \({\bar{h}} \in {\mathfrak {D}}({{\mathbf {U}}(0,1+\iota )})\) and \({\bar{h}} [S ]\subseteq R + {\mathbf {B}}(0,2^{-12}\varepsilon )\). To map S onto R we shall compose \({\bar{h}}\) with the following map
where \(\lambda : {\mathbf {R}}\rightarrow {\mathbf {R}}\) is a \({\mathscr {C}}^\infty \) smooth map such that
Noting that \(|({T}_\natural x)/|{T}_\natural x| - x| = {{\mathrm{dist}}}(x,R)\) we see that \({{\mathrm{Lip}}}k\) does not depend on \(\varepsilon \). Therefore we may set \(h = k \circ {\bar{h}}\). \(\square \)
Remark 11.6
Note that if S was allowed to approach \({{\mathrm{Bdry}}}{\mathbf {B}}(0,1)\) tangentially, i.e., if we did not assume \(S \cap (R + {\mathbf {B}}(0,\delta )) \subseteq T\), then the auxiliary map \({\bar{g}}\) constructed in the proof above might have not been Lipschitz continuous.
In the next lemma we basically prove that the \({\mathscr {H}}^m\) measure of the unrectifiable part of elements of any minimising sequence must vanish in the limit.
Lemma 11.7
Assume
Denoting the purely \(({\mathscr {H}}^m,m)\) unrectifiable part of \(S_i \cap U\) by \(\bar{S}_i\) we have
Proof
Let \(x_0 \in U\) be such that \(\varvec{\Theta }^m(\Vert V\Vert ,x_0) \in (0,\infty )\)–from 11.3 we know that \(\Vert V\Vert \) almost all \(x_0\) satisfy this condition. Without loss of generality we shall assume \(x_0 = 0\). Suppose (120) is not true at \(x_0\). Then there exists a subsequence \(\{ R_i : i \in {\mathscr {P}}\}\) of \(\{ S_i : i \in {\mathscr {P}}\}\), a sequence of radii \(\{ r_j : j \in {\mathscr {P}}\}\) with \(r_j \downarrow 0\) as \(j \rightarrow \infty \), and \(\delta \in (0,\infty )\) such that denoting \(R_{j,i} = \varvec{\mu }_{1/r_j}[R_i ]\cap {\mathbf {B}}(0,1)\) we get
where \({\bar{R}}_{j,i}\) denotes the purely \(({\mathscr {H}}^m,m)\) unrectifiable part of \(R_{j,i}\). Set \({{\hat{R}}}_{j,i} = R_{j,i} {{\mathrm{\sim }}}{\bar{R}}_{j,i}\). Using [19, 2.9.11, 2.8.18] we may and shall assume that
Set \(\xi _1 = \inf {{\mathrm{im}}}F\) and \(\xi _2 = \sup {{\mathrm{im}}}F\) and let \(\varvec{\beta }(n)\) be the optimal constant from the Besicovitch–Federer covering theorem [19, 2.8.14]. Choose \(\iota \in (0,2^{-144})\) so that
Define
For each \(i,j \in {\mathscr {P}}\) with \(i \ge i_0(j)\) employ the Besicovitch-Federer covering theorem [19, 2.8.14] to obtain at most countable subfamily \({\mathcal {B}}_{j,i}\) of \(\bar{{\mathcal {B}}}_{j,i}\) such that
Define \(E_{j,i} = \mathop {{\textstyle \bigcup }}{\mathcal {B}}_{j,i} \cap {\mathbf {B}}(0,1)\) and observe that
Employ 10.4 with \(\iota \) in place of \(\varepsilon \) to obtain the map \(f_{j,i} \in {\mathfrak {D}}({{{\mathrm{Int}}}E_{j,i}})\) of class \({\mathscr {C}}^{\infty }\) such that
Observe that we have
hence, we choose \(j,i_1 \in {\mathscr {P}}\) such that \(i_i \ge i_0(j)\) and
In consequence, recalling (123) and assuming (122) we obtain for all \(j \in {\mathscr {P}}\)
This contradicts the definition of \(\mu \) so (122) cannot hold and (120) is proven.
We will now show the second claim of the lemma. Assume (121) is not true. Then there exists a subsequence \(\{ Q_i : i \in {\mathscr {P}}\}\) of \(\{S_i : i \in {\mathscr {P}}\}\) and \(\vartheta \in (0,\infty )\) such that
where \({\bar{Q}}_i\) denotes the purely \(({\mathscr {H}}^m,m)\) unrectifiable part of \(Q_i \cap U\). Let \(\{\nu _i : i \in {\mathscr {P}}\}\) be a convergent subsequence of 
. Observe that (124) implies that \(\nu = \lim _{i \rightarrow \infty } \nu _i\) is a finite non-zero Radon measure over U. Recalling (120) we see that
Moreover, \(\nu \) is absolutely continuous with respect to \(\Vert V\Vert \) which, employing 9.4, is absolutely continuous with respect to 
. Thus, recalling (1) and 11.3 , and using [19, 2.9.7, 2.8.18, 2.9.5], we deduce that
Hence, \(\nu \) could not be non-zero and (121) is proven. \(\square \)
Theorem 11.8
Assume U, F, \({\mathcal {C}}\), \(\{ S_i : i \in {\mathscr {P}}\} \subseteq {\mathcal {C}}\), V, \(\mu \) are as in 11.2 and
Then
-
(a)
\(\varvec{\Theta }^m(\Vert V\Vert ,x_0) \ge 1\).
Moreover, if F is elliptic, then
-
(b)
\(\varvec{\Theta }^m(\Vert V\Vert ,x_0) = 1\);
-
(c)
\({{\mathrm{VarTan}}}(V,x_0) = \{ {\mathbf {v}}_{m}(T) \}\).
Proof
Proof of (a). Define \(E = {{\mathrm{spt}}}\Vert V\Vert \) and \(B = {\mathbf {R}}^n {{\mathrm{\sim }}}U\). Without loss of generality, we assume \(x_0 = 0\). Employing 11.2 we shall also assume that \(\{S_i : i \in {\mathscr {P}}\}\) satisfies all the conclusions of 11.2. In particular, for some \(S \in {\mathcal {C}}\) and all compacts sets \(K \subseteq U\)
Define
Recall [1, 3.4(1)] and \(\varvec{\Theta }^m(\Vert V\Vert ,x_0) \in (0,\infty )\) to see that \({{\mathrm{VarTan}}}(V,0)\) is compact and nonempty so we can choose \(C \in {{\mathrm{VarTan}}}(V,0)\) and \(\{ r_j \in {\mathbf {R}}: j \in {\mathscr {P}}\}\) such that \(r_j \downarrow 0\) as \(j \rightarrow \infty \), and \(\delta (r_1) < 1\), and \({\mathbf {U}}(0,3r_1)\subseteq U\), and
Set \(\delta _j = \delta (r_j)\) and \(\varepsilon _j = 10 \delta _j^{1/2}\). For \(j \in {\mathscr {P}}\) let \(f_j, g_j, h_j \in {\mathscr {C}}^{\infty }({\mathbf {R}},{\mathbf {R}})\) be such that
We define \(p_j \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n,{\mathbf {R}}^n)\) and \(q_j \in {\mathscr {C}}^{\infty }({\mathbf {R}}^n,{\mathbf {R}}^n)\) so that
We set \(B_j = {\mathbf {U}}(0,2r_j) \cap \bigl ( T + {\mathbf {B}}(0,2\delta _jr_j) \bigr )\). Then
Using (125) and possibly passing to a subsequence of \(\{ S_i : i \in {\mathscr {P}}\}\) we shall further assume that for \(i,j \in {\mathscr {P}}\) with \(i \ge j\) there holds
For \(j \in {\mathscr {P}}\) the map \(p_j\) is clearly admissible so for \(i \in {\mathscr {P}}\) we have \(p_j [S_i ]\in {\mathcal {C}}\) and
Define \(A_j = {\mathbf {B}}(0,r_j) {{\mathrm{\sim }}}{\mathbf {U}}(0,(1-3\varepsilon _j)r_j)\) for \(j \in {\mathscr {P}}\) and \(\xi _1 = \inf {{\mathrm{im}}}F > 0\) and \(\xi _2 = \sup {{\mathrm{im}}}F < \infty \). For \(i,j \in {\mathscr {P}}\) with \(i \ge j\), recalling (126), we have
where \(\kappa _j = \xi _2 \big ( 6 + \delta _j^{1/2} \big )^m - \xi _1\) and \(\kappa _{\infty } = 6^m\xi _2 - \xi _1\). In consequence
Since \({\mathbf {v}}_{m}(S_i \cap U) \rightarrow V \in {\mathbf {V}}_{m}(U)\) as \(i \rightarrow \infty \) and \(A_j\) is compact, we have
Since \((1-\varepsilon _j/2)r_j+3\delta _jr_j< r_j\), we get
so we obtain for \(i,j \in {\mathscr {P}}\) with \(i \ge j\)
Hence, recalling (128) and (129), we see that for \(j \in {\mathscr {P}}\)
For \(i,j \in {\mathscr {P}}\) define, recalling 3.7,
Then
Since \({\mathbf {v}}_{m}(\varvec{\mu }_{1/r_j} [S_i ]) = (\varvec{\mu }_{1/r_j})_{\#} {\mathbf {v}}_{m}(S_i)\) by 3.14, we get for \(j \in {\mathscr {P}}\) using (130)
hence, recalling \(\varvec{\Theta }^m(\Vert V\Vert ,x_0) \in (0,\infty )\) and [1, 3.4(2)],
Employing (127) we obtain for \(i,j \in {\mathscr {P}}\) with \(i \ge j\)
For \(i,j \in {\mathscr {P}}\) define
Since \(W_{j,i}\) is closed we have \(Y_{j,i} \subseteq W_{j,i} \cap {\mathbf {B}}(0,\rho _j)\) and \({\mathscr {H}}^m({\mathbf {B}}(0,\rho _j) \cap W_{j,i} {{\mathrm{\sim }}}Y_{j,i}) = 0\). Roughly speaking, \(Y_{j,i}\) equals \(W_{j,i} \cap {\mathbf {B}}(0,\rho _j)\) with removed “hair”. We will now check that for \(i,j \in {\mathscr {P}}\) big enough with \(i \ge j\), one cannot deform \(Y_{j,i}\) onto \(R_j = T \cap {{\mathrm{Bdry}}}{\mathbf {B}}(0,\rho _j)\) by any Lipschitz continuous map \({\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) which fixes \(R_j\). Assume, by contradiction, that there exists such a retraction \({\bar{\phi }}_j\). Observe that whenever \(\gamma \in (0,\varepsilon _j/2)\)
For each \(i,j \in {\mathscr {P}}\) we choose \(\gamma _{j,i} \in (0,\varepsilon _j/2)\) such that
Recalling (127) and that \(\varepsilon _j/2 = 5 \delta _j^{1/2}\) we see that
Hence, by Lemma 11.5, there exists \(\phi _j \in {\mathfrak {D}}({{\mathbf {U}}(0,\rho _j + \gamma _{j,i})})\) such that \(\phi _j [Y_{j,i} ]\subseteq R_j\) and \({{\mathrm{Lip}}}\phi _j \le \Gamma _{11.5}({{\mathrm{Lip}}}{\bar{\phi }}_j, \varepsilon _j/2)\). Clearly \(\varvec{\mu }_{r_j} \circ \phi _j[W_{j,i}] \in {\mathcal {C}}\) so using (132) we get
We choose \(j_0 \in {\mathscr {P}}\) so big that for \(j \ge j_0\) we have
which is possible because
For each \(j \ge j_0\) we select \(i_0 = i_0(j) \in {\mathscr {P}}\) such that \(i_0 \ge j\) and for \(i \ge i_0\)
which is possible because \(\{ S_i : i \in {\mathscr {P}}\}\) is a minimising sequence and due to (129). Combining (133), (134), and (135) we get for \(j \ge j_0\) and \(i \ge i_0(j)\) the following contradictory estimate
We now know that \(W_{j,i}\) cannot be deformed onto \(R_j = T \cap {{\mathrm{Bdry}}}{\mathbf {B}}(0,\rho _j)\) by any Lipschitz continuous map \({\mathbf {R}}^n \rightarrow {\mathbf {R}}^n\) fixing \(R_j\). In consequence we get
because otherwise we could deform \(W_{j,i}\) onto \(R_j\). In particular,
so (a) is now proven.
Proof of (b). Define \(X_j = {\mathbf {U}}(0,1) {{\mathrm{\sim }}}{\mathbf {B}}(0,\rho _j)\) and let \(F^0\) be defined as in 3.8. Recall (131) and (137). Note that whenever \(A \subseteq R^n\) is closed and satisfies \({\mathscr {H}}^m(A \cap K) < \infty \) for all compact \(K \subseteq {\mathbf {R}}^n\), and \(j \in {\mathscr {P}}\), then
Since F is elliptic and \(\varvec{\mu }_{1/\rho _j} [Y_{j,i} ]\) satisfies 3.16(b) employing (127) we can find \(\xi _3 > 0\) such that for \(i,j \in {\mathscr {P}}\) with \(j \ge j_0\) and \(i \ge i_0(j)\)
Since \({\widetilde{q}}_j( {\mathbf {U}}(0,1) ) \subseteq {\mathbf {U}}(0,1)\) and \({\widetilde{q}}_j(x) = x\) for \(x \in {\mathbf {U}}(0,1-3\varepsilon _j) \cup ({\mathbf {R}}^n {{\mathrm{\sim }}}{\mathbf {U}}(0,1))\), we see that
Hence, we get
Recalling (126) and 3.15 we see that
We observe that
Let us define \(\omega : (0,\infty ) \rightarrow {\mathbf {R}}\) by the formula
Then, we may write
Using again (126) we have
Since F is continuous and \({\mathbf {G}}(n,m)\) is compact we see that \(\lim _{r \rightarrow 0} \omega (r) = 0\); hence, combining (138) with (139), (140), (141), (142), and (143) we obtain
For any \(i,j \in {\mathscr {P}}\) we have
Since \(V = \lim _{i \rightarrow \infty } {\mathbf {v}}_{m}(S_i \cap U)\) is minimising, we obtain
hence, transforming (145) we get
Estimating as in (139), (140), (141) and using (121) we reach the conclusion
Plugging this into (144) we obtain
Hence, 
by [1, 3.4(2)]. To have \(C = {\mathbf {v}}_{m}(T)\) we still need to show that \(P = T\) for C almost all (x, P).
Proof of (c). Let \({{\hat{S}}}_{j,i}\) denote the \(({\mathscr {H}}^m,m)\) rectifiable part of \(S_{j,i} \cap {\mathbf {B}}(0,1)\). From 11.7 and (b) we know that
Set \(p_{j,i} = {T}_\natural |_{{{\hat{S}}}_{j,i}}\). Since \(p_{j.i}(x) = \widetilde{p}_j(x)\) for \(x \in S_{j,i} \cap {\mathbf {B}}(0,1) {{\mathrm{\sim }}}{{\widetilde{A}}}_j\) and recalling (141) and (136) we see that
Clearly \({{\mathrm{Lip}}}p_{j,i} \le 1\) so \(1 - {{\mathrm{ap}}}J_m p_{j,i}(x) \ge 0\) for \({\mathscr {H}}^m\) almost all \(x \in {{\hat{S}}}_{j,i}\). Employing the area formula [19, 3.2.20] we have
Hence, employing 11.4, we get
Define \(\varphi (x,P) = \mathbb {1}_{{\mathbf {B}}(0,1)}(x) \Vert {P}_\natural - {T}_\natural \Vert ^2\) for \((x,P) \in {\mathbf {R}}^n \times {\mathbf {G}}(n,m)\). Recalling 11.7 and noting that \(\Vert C\Vert ({{\mathrm{Bdry}}}{\mathbf {B}}(0,1)) = 0\), we see that
Therefore, \(P = T\) for C almost all \((x,P) \in {\mathbf {B}}(0,1) \times {\mathbf {G}}(n,m)\).
Now, since \(C \in {{\mathrm{VarTan}}}(V,0)\) was chosen arbitrarily we obtain for all \(C \in {{\mathrm{VarTan}}}(V,0)\)
In particular, if \(C \in {{\mathrm{VarTan}}}(V,0)\) and \(\rho \in (0,1)\), then \(C' = (\varvec{\mu }_{\rho })_{\#}C \in {{\mathrm{VarTan}}}(V,0)\) also satisfies (147); hence, for all \(C \in {{\mathrm{VarTan}}}(V,0)\)
Now we have all the ingredients to prove our main theorem.
Proof of 3.20
First we recall 11.3 too obtain S and \(\{ S_i : i \in {\mathscr {P}}\}\) and too see that we may apply 11.8 at \(\Vert V\Vert \) almost all \(x_0\). We then get \(\varvec{\Theta }^m(\Vert V\Vert ,x) = 1\) and \({{\mathrm{Tan}}}({{\mathrm{spt}}}\Vert V\Vert ,x) = T\) for V almost all \((x,T) \in U \times {\mathbf {G}}(n,m)\). We know \({\mathscr {H}}^m(S \cap U {{\mathrm{\sim }}}{{\mathrm{spt}}}\Vert V\Vert ) = 0\), which means that \(V = {\mathbf {v}}_{m}(S \cap U)\) and that S and \(\{ S_i : i \in {\mathscr {P}}\}\) satisfy all the conditions of 3.20. \(\square \)
12 An example of a good class: homological spanning
Let us fix an abelian coefficient group G. We shall work in the category \({\mathcal {A}}_1\) of arbitrary pairs and their maps as defined in [16, I,1]. This means that (X, A) is an object in \({\mathcal {A}}_1\) if X is a topological space and \(A \subseteq X\) is an arbitrary subset with the relative topology. Morphisms in \({\mathcal {A}}_1\) between (X, A) and (Y, B) are continuous functions \(f : X \rightarrow Y\) such that \(f [A ]\subseteq B\). If (X, A), (Y, B) are objects in \({\mathcal {A}}_1\) and \(f : (X,A) \rightarrow (Y,B)\) is a morphism in \({\mathcal {A}}_1\), then the symbol \({\check{\mathbf {H}}}_k(X,A;G)\) shall denote the \(k^{\mathrm {th}}\) Čech homology group [16, IX,3.3] of the pair (X, A) with coefficients in G and \({\check{\mathbf {H}}}_k(f) : {\check{\mathbf {H}}}_k(X,A;G) \rightarrow {\check{\mathbf {H}}}_k(Y,B;G)\) the induced morphism of abelian groups. In case \(A = \varnothing \), we write \({\check{\mathbf {H}}}_k(X;G) = {\check{\mathbf {H}}}_k(X,\varnothing ;G)\). For any sets \(X \subseteq Y \subseteq {\mathbf {R}}^n\), we will denote by \(i_{X,Y}\) the inclusion map \(X \hookrightarrow Y\).
Definition 12.1
Let B be a closed subset of \({\mathbf {R}}^{n}\), L a subgroup of \({\check{\mathbf {H}}}_{m-1}(B;G)\). We say that a closed set \(E \subseteq {\mathbf {R}}^n\) spans L if \(L \subseteq \ker \bigl ( {\check{\mathbf {H}}}_{m-1}(i_{B, B\cup E}) \bigr )\). In other words, E spans L if the composition
is zero.
We denote by \(\check{{\mathcal {C}}}(B,L,G)\) the collection of all closed subsets of \({\mathbf {R}}^n\) which span L.
We shall prove that \(\check{{\mathcal {C}}}(B,L,G)\) is a good class in the sense of definition 3.4.
Lemma 12.2
Let B be a closed subset of \({\mathbf {R}}^n\), L a subgroup of \({\check{\mathbf {H}}}_{m-1}(B;G)\). Let \(\{ E_{k} \subseteq {\mathbf {R}}^n : k \in {\mathscr {P}}\}\) be a decreasing sequence of closed sets. If \(B \subseteq E_{k+1} \subseteq E_k\) and \(L \subseteq \ker \bigl ( {\check{\mathbf {H}}}_{m-1}(i_{B,E_{k}}) \bigr )\) for all \(k \ge 1\), then, by setting \(E = \bigcap _{k \ge 1} E_k\), we have \(L \subseteq \ker \bigl ( {\check{\mathbf {H}}}_{m-1}(i_{B,E}) \bigr )\).
Proof
Since \(E = \bigcap _{k\ge 1} E_k\), we have that \(E=\varprojlim E_k\), see for example [16, Theorem 2.5 on p. 260]. We let
be the natural isomorphism, and let
be the natural projections. Then
Since
by the universal property of inverse limit, there exist a homomorphism
such that
Then
thus \(\psi (L)=0\). We see that
thus
Remark 12.3
For any continuous map \(\varphi :{\mathbf {R}}^{n}\rightarrow {\mathbf {R}}^{n}\) with \(\varphi \vert _{B}=\mathrm {id}_{B}\), we have
In particular, \(\check{{\mathcal {C}}}(B,L,G)\) satisfies condition (c) of Definition 3.4.
Lemma 12.4
Let B, L and \(\check{{\mathcal {C}}}(B,L,G)\) be given as in Definition 12.1. Then \(\check{{\mathcal {C}}}(B,L,G)\) is a good class in the sense of 3.4.
Proof
Observe that \({\mathbf {R}}^n \in \check{{\mathcal {C}}}(B,L,G)\) so \(\check{{\mathcal {C}}}(B,L,G)\) is non-empty and contains only closed sets by definition. Recalling 12.3 we only need to check condition (d) of Definition 3.4.
If \(\{ S_k:k\in {\mathscr {P}}\}\in \check{{\mathcal {C}}}(B,L,G)\) is a sequence such that \(S_i \rightarrow S\) locally in Hausdorff distance for some closed set S, then by putting
we have that
By Lemma 12.2, we get that \(L\subseteq \ker {\check{\mathbf {H}}}_{m-1}\left( i_{B,B\cup S} \right) \). \(\square \)
Remark 12.5
Replacing Čech homology group with Čech cohomology group, we let L be a subgroup of \({\check{\mathbf {H}}}^{m-1}(B;G)\), \({\mathcal {C}}\) a collection of closed sets E such that the composition
is zero. By continuity of Čech cohomology theory, see for example [16, Theorem 3.1 in p. 261], we get also that \({\mathcal {C}}\) is a good class. Indeed, we have a similar result as Lemma 12.2, but with Čech cohomology groups instead of Čech homology groups. For the proof we refer the reader to [39, Proposition (2.7)].
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Acknowledgements
Both authors acknowledge the perfect working conditions they had while working at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Potsdam-Golm. Part of this work has been done while the authors were visiting Xiangyu Liang at the Institut Camille Jordan, Université Claude Bernard Lyon 1. We acknowledge her kind hospitality. The authors are also thankful to Ulrich Menne for giving the incentive to start this project, guidance through intricate paths of Almgren’s brilliant ideas, and constant support along the way. The second author was partially supported by NCN Grant No. 2013/10/M/ST1/00416 Geometric curvature energies for subsets of the Euclidean space.
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Fang, Y., Kolasiński, S. Existence of solutions to a general geometric elliptic variational problem. Calc. Var. 57, 91 (2018). https://doi.org/10.1007/s00526-018-1348-4
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DOI: https://doi.org/10.1007/s00526-018-1348-4