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

By Martin Kell, Andrea Mondino. 2016

86 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/1a70a14633551398d697661e8fd5d927baf9877e

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In Applied Mathematics

Citations in the corpus

Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “This follows for instance from the rectifiability results proved in [40], [37], [31] (see Theorem 2.”
On a conjecture of Cheeger (2016, result): “After we finished writing this note we learned that similar results have been proved by Kell and Mondino [KM16] and by Gigli and Pasqualetto [GP16].”
Dimensional estimates and rectifiability for measures satisfying linear PDE constraints (2018, background): “heorem [1] (see also [24] for a diﬀerent proof based on a geometrical argument). • If A = div, combining Theorem 1.1 with the result of [2], one obtains the weak converse of Rademacher’s theorem (see [14, 19, 21] for other consequences in metric geometry). The main results of this paper is to show h”
Behaviour of the reference measure on $\sf RCD$ spaces under charts (2016, result): “Finally we remark that other two independent recent papers ([7] and [14]) cover results overlapping with ours; let us briefly describe those and the relations with ours.”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “After [42, 38, 21, 32] we know that they are rectifiable as metric measure spaces.”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “See [17,35] and [27] for the proof of the absolute continuity statement.”