Connections between metric differentiability and rectifiability

Caamaño, Iván; Durand-Cartagena, Estibalitz; Jaramillo, Jesús; Prieto, Ángeles; Soultanis, Elefterios

doi:10.1090/proc/17123

Connections between metric differentiability and rectifiability
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by Iván Caamaño, Estibalitz Durand-Cartagena, Jesús Á. Jaramillo, Ángeles Prieto and Elefterios Soultanis;

Proc. Amer. Math. Soc. 153 (2025), 2075-2088

DOI: https://doi.org/10.1090/proc/17123

Published electronically: February 20, 2025

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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.

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