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Connections between metric differentiability and rectifiability

Iván Caamaño, Estíbalitz Durand-Cartagena, Jesús Á. Jaramillo, Ángeles Prieto, Elefterios Soultanis

Abstract

We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection $\mathcal C$ of Banach (or metric) spaces: if a metric measure space $X$ bi-Lipschitz embeds in some element in $\mathcal C$, and if every Lipschitz map $X\to Y\in \mathcal C$ is differentiable, then $X$ is rectifiable. This gives a simple proof of the rectifiability of Lipschitz differentiability spaces that are bi-Lipschitz embeddable in Euclidean space, due to Kell--Mondino. Our principle also implies a converse to Kirchheim's theorem: if all Lipschitz maps from a domain space to arbitrary targets are metrically differentiable, the domain is rectifiable. We moreover establish the compatibility of metric and w$^*$-differentials of maps from metric spaces in the spirit of Ambrosio--Kirchheim.

Connections between metric differentiability and rectifiability

Abstract

We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection of Banach (or metric) spaces: if a metric measure space bi-Lipschitz embeds in some element in , and if every Lipschitz map is differentiable, then is rectifiable. This gives a simple proof of the rectifiability of Lipschitz differentiability spaces that are bi-Lipschitz embeddable in Euclidean space, due to Kell--Mondino. Our principle also implies a converse to Kirchheim's theorem: if all Lipschitz maps from a domain space to arbitrary targets are metrically differentiable, the domain is rectifiable. We moreover establish the compatibility of metric and w-differentials of maps from metric spaces in the spirit of Ambrosio--Kirchheim.
Paper Structure (5 sections, 8 theorems, 42 equations)

This paper contains 5 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric and linear differentials
  4. Metric differential and rectifiability
  5. Metric and $w^*$-differentials

Key Result

Theorem 1.2

Let $(U,\varphi)$ be an $n$-dimensional chart in $X$ satisfying eq:lin-uniqueness, and $\mathcal{C}$ a collection of metric spaces so that some $Y\in \mathcal{C}$ contains a non-trivial geodesic. If every Lipschitz map $U\to Y\in\mathcal{C}$ admits a metric differential with respect to $(U,\varphi)$

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Corollary 3.2
  • proof : Proof of Proposition \ref{['prop:kir-implies-che']}
  • Lemma 4.1
  • ...and 8 more