A compactness result for Fano manifolds and Kahler Ricci flows

We obtain a compactness result for Fano manifolds and Kahler 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 Kahler 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 Kahler Ricci flows. The proof is based on a Holder estimate for the gradient of harmonic functions and mixed derivative of Green's function.NSF; Siyuan Foundation; Simons FoundationSCI(E)ARTICLEtian@math.princeton.edu; qizhang@math.ucr.edu3-4965-99936