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Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows
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We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K,N) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting.
Forward citations
Cited by 3 Pith papers
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On the structure of RCD spaces with upper curvature bounds
Every RCD space with curvature bounded above is a topological manifold with boundary whose interior is the regular set, a smooth geodesically convex manifold.
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Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces
In RCD(K,N) spaces, the reduced boundary of a set of finite perimeter has a unique Euclidean half-space tangent at almost every point and is rectifiable by bi-Lipschitz charts.
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On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces
The author proves W-entropy dissipation and Shannon entropy power concavity on closed (K,n,N)-super Ricci flows over metric measure spaces, and connects lower-bounded W-entropy to volume non-collapsing on RCD(0,N) spaces.
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