Connections between metric differentiability and rectifiability
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arXiv
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| Format: | Preprint |
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2024
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| author | Caamaño, Iván Durand-Cartagena, Estíbalitz Jaramillo, Jesús Á. Prieto, Ángeles Soultanis, Elefterios |
| author_facet | Caamaño, Iván Durand-Cartagena, Estíbalitz Jaramillo, Jesús Á. Prieto, Ángeles Soultanis, Elefterios |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18440 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connections between metric differentiability and rectifiability Caamaño, Iván Durand-Cartagena, Estíbalitz Jaramillo, Jesús Á. Prieto, Ángeles Soultanis, Elefterios Metric Geometry 30L05, 30L99, 51F30 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. |
| title | Connections between metric differentiability and rectifiability |
| topic | Metric Geometry 30L05, 30L99, 51F30 |
| url | https://arxiv.org/abs/2403.18440 |