Calculus of Variations and Geometric Measure Theory

M. Kell - A. Mondino

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

created by mondino on 08 Jul 2016
modified on 20 Jan 2017


Accepted Paper

Inserted: 8 jul 2016
Last Updated: 20 jan 2017

Journal: Annali SNS
Year: 2016
Doi: 10.2422/2036-2145.201608_007


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.