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.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2019
- DOI:
- 10.48550/arXiv.1905.00123
- arXiv:
- arXiv:1905.00123
- Bibcode:
- 2019arXiv190500123H
- Keywords:
-
- Mathematics - Differential Geometry;
- Mathematics - Metric Geometry
- E-Print:
- 18 pages. To appear in Geometry &