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A compactness result for Fano manifolds and Kähler Ricci flows

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Abstract

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 \(C^1\) 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.

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Acknowledgments

Q. S. Z. would like to thank Professors X. X. Chen, G. F. Wei and Zhenlei Zhang for helpful conversations. We are also grateful to the referee for helpful comments.

   After the paper have been accepted for publication, Professor Xiaohua Zhu kindly informed us that the Proof of Theorems 1.2 and 1.3 can be shortened by proving \(C^{1, \alpha }\) continuity of harmonic functions and the Ricci potential within a harmonic coordinate by standard Schauder method. Thus the two theorems can also be strengthened to allow any \(\alpha \in (0, 1)\) instead of some \(\alpha \in (0, 1)\). However the integral estimate in Sect. 2 may be of independent interest.

   G. T. acknowledges the support of a NSF grant. Part of the paper was written when Q. S. Z. was a visiting professor of Nanjing University under a Siyuan Foundation grant. He is grateful to both the Siyuan Foundation and the Simons Foundation for their support.

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Authors and Affiliations

  1. BICMR, Peking University, Beijing, 100871, China

    Gang Tian

  2. Department of Mathematics, Princeton University, Princeton, NJ, 02139, USA

    Gang Tian

  3. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Qi S. Zhang

  4. Department of Mathematics, University of California, Riverside, CA, 92521, USA

    Qi S. Zhang

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  1. Gang Tian
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Correspondence to Qi S. Zhang.

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Tian, G., Zhang, Q.S. A compactness result for Fano manifolds and Kähler Ricci flows. Math. Ann. 362, 965–999 (2015). https://doi.org/10.1007/s00208-014-1147-y

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  • DOI: https://doi.org/10.1007/s00208-014-1147-y

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