New differential operator and non-collapsed $RCD$ spaces

Abstract

We show characterizations of non-collapsed compact $RCD(K,N)$ spaces, which in particular confirm a conjecture of De Philippis-Gigli on the implication from the weakly non-collapsed condition to the non-collapsed one in the compact case. The key idea is to give the explicit formula of the Laplacian associated to the pull-back Riemannian metric by embedding in $L^2$ via the heat kernel. This seems the first application of geometric flow to the study of RCD spaces.

arXiv comment18 pages. To appear in Geometry & Topology

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Cited by in corpus (10)

• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• Singular Weyl's law with Ricci curvature bounded below
• An upper bound on the revised first Betti number and a torus stability result for RCD spaces
• On the structure of RCD spaces with upper curvature bounds
• Weakly non-collapsed RCD spaces are strongly non-collapsed
• On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited
• The metric measure boundary of spaces with Ricci curvature bounded below
• Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces
• A characterization of non-collapsed $RCD(K,N)$ spaces via Einstein tensors
• Sphere theorems for RCD and stratified spaces