New differential operator and noncollapsed RCD spaces

Honda, Shouhei

doi:10.2140/gt.2020.24.2127

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New differential operator and noncollapsed $\mathrm{RCD}$ spaces

Shouhei Honda

Geometry & Topology 24 (2020) 2127–2148

DOI: 10.2140/gt.2020.24.2127

Abstract

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

Dedicated to Professor Kenji Fukaya on his 60th birthday

Keywords

Ricci curvature, Laplacian, metric measure space

Mathematical Subject Classification 2010

Primary: 53C21

References

Publication

Received: 28 June 2019

Revised: 18 August 2019

Accepted: 23 September 2019

Published: 10 November 2020

Proposed: Tobias H Colding

Seconded: Gang Tian, Bruce Kleiner

Authors

Shouhei Honda
Mathematical Institute Tohoku University Sendai Japan

Volume 24, issue 4 (2020)