  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.
