On the volume measure of non-smooth spaces with Ricci curvature bounded below

TL;DR

This paper establishes a measure-theoretic decomposition for non-smooth spaces with Ricci curvature bounds, showing they can be covered by subsets with well-understood dimensional measures, and proves rectifiability results for Lipschitz differentiability spaces.

Contribution

It introduces a measure decomposition for $RCD^{*}(K,N)$ spaces and proves rectifiability of Lipschitz differentiability spaces embedded in Euclidean space.

Findings

Spaces can be covered by subsets with measures absolutely continuous to Hausdorff measures of varying dimensions.

Lipschitz differentiability spaces embedded in Euclidean space are rectifiable.

Application to Alexandrov spaces demonstrates the broader relevance.

Abstract

We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces.