A compactness result for Fano manifolds and K\"ahler Ricci flows

We obtain a compactness result for Fano manifolds and K\"ahler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the K\"ahler Ricci flows. One assumption is the boundedness of the Ricci potential and the other is the smallness of Perelman's entropy. As one application, we obtain a new local regularity criteria and structure result for K\"ahler Ricci flows. The proof is based on a H\"older estimate for the gradient of harmonic functions, which may be of independent interest

Isoperimetric inequality under K\"ahler Ricci flow

Let ({\M}, g(t)) be a K\"ahler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on ({\M}, g(t)), and a Poincar\'e inequality without assuming the Ricci curvature is bounded from below.Comment: final version, to appear in Am. J. Mat