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Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space

Sean Li, Raanan Schul

TL;DR

This work characterizes $p$-rectifiability of a compact metric space $X$ with $0<\mathcal{H}^p(X)<\infty$ and positive lower density by a biLipschitz-decomposition property: $X$ is $p$-rectifiable if and only if every Lipschitz map on a subset $F\subset X$ with positive $\mathcal{H}^p$-image has a positive-measure subset on which the map is biLipschitz, equivalently admitting a full biLipschitz decomposition. The authors develop independent Alberti representations and a constructive method to extend them, build a metric $d$ via a hierarchical system of shortcuts, and prove David–Semmes regularity to compare $d$ with the original metric $\rho$. A central innovation is a converse to the Sard-like decomposition: if biLipschitz pieces always exist, the space must be rectifiable; otherwise one can produce a Lipschitz map with positive image but no biLipschitz piece on any positive-measure subset, yielding a new Euclidean-variant characterization of unrectifiability through transversal curve fragments. The results provide new tools for analyzing rectifiability, including a method to generate independent Alberti representations and a framework that links geometric measure theory with metric-geometry constructions such as DS-regularity and minimal looking-down questions.

Abstract

We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset $F$ and $f:F\to Y$, a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset $A\subset F$ so that $f$ is biLipschitz on $A$. We also give a characterization in terms of a full biLipschitz decomposition. These characterizations are new even for subsets of Euclidean space. One of our tools is Alberti representations. On the way we give a method for constructing independent Alberti representations, which may be of independent interest. We use this to characterize unrectifiable metric spaces as those spaces for which there exist a positive measure subset $S$ and a Lipschitz map $φ$ into a lower dimensional Euclidean space so that $S$ is $\cH^1$-null with respect to all curve fragments that are quantitatively transversal to $φ$.

Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space

TL;DR

This work characterizes -rectifiability of a compact metric space with and positive lower density by a biLipschitz-decomposition property: is -rectifiable if and only if every Lipschitz map on a subset with positive -image has a positive-measure subset on which the map is biLipschitz, equivalently admitting a full biLipschitz decomposition. The authors develop independent Alberti representations and a constructive method to extend them, build a metric via a hierarchical system of shortcuts, and prove David–Semmes regularity to compare with the original metric . A central innovation is a converse to the Sard-like decomposition: if biLipschitz pieces always exist, the space must be rectifiable; otherwise one can produce a Lipschitz map with positive image but no biLipschitz piece on any positive-measure subset, yielding a new Euclidean-variant characterization of unrectifiability through transversal curve fragments. The results provide new tools for analyzing rectifiability, including a method to generate independent Alberti representations and a framework that links geometric measure theory with metric-geometry constructions such as DS-regularity and minimal looking-down questions.

Abstract

We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset and , a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset so that is biLipschitz on . We also give a characterization in terms of a full biLipschitz decomposition. These characterizations are new even for subsets of Euclidean space. One of our tools is Alberti representations. On the way we give a method for constructing independent Alberti representations, which may be of independent interest. We use this to characterize unrectifiable metric spaces as those spaces for which there exist a positive measure subset and a Lipschitz map into a lower dimensional Euclidean space so that is -null with respect to all curve fragments that are quantitatively transversal to .
Paper Structure (9 sections, 30 theorems, 142 equations)

This paper contains 9 sections, 30 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Alberti representations
  4. Constructing independent Alberti representations
  5. Lower bounding itineraries
  6. Constructing shortcuts
  7. David-Semmes regularity
  8. Proof of Lemma \ref{['l:base']}
  9. Proof of Theorem \ref{['th:main']}

Key Result

Lemma 1.1

Let $f : A \to Y$ be Lipschitz where $A \subseteq \mathbb{R}^p$ is Borel and $Y$ is a metric space. Then there exist a Borel decomposition so that $\mathcal{H}^p(f(N)) = 0$ and $f|_{E_i}$ is biLipschitz for each $i$.

Theorems & Definitions (59)

  • Lemma 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof : Proof of Corollary \ref{['c:cor2']}
  • Corollary 1.4
  • proof
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • ...and 49 more