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

Shouhei Honda

Geometry & Topology 24 (2020) 2127–2148

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 L2 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

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Ricci curvature, Laplacian, metric measure space
Mathematical Subject Classification 2010
Primary: 53C21
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
Shouhei Honda
Mathematical Institute
Tohoku University