A compactness result for Fano manifolds and Kähler Ricci flows

We obtain a compactness result for Fano manifolds and Kähler Ricci flows. Comparing to the more general Riemannian versions in Anderson (Invent Math 102(2):429–445, 1990) and Hamilton (Am J Math 117:545–572, 1995), in this Fano case, the curvature assumption is much weaker and is preserved by the Kähler Ricci flows. One assumption is the $$C^1$$^C1 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 Kähler Ricci flows. The proof is based on a Hölder estimate for the gradient of harmonic functions and mixed derivative of Green’s function.