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

Abstract

We show characterizations of noncollapsed compact $RCD\left(K,N\right)$ 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.