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Regularity Scales and Convergence of the Calabi Flow

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A Correction to this article was published on 26 July 2018

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Abstract

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kähler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.

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Change history

  • 26 July 2018

    The authors would like to acknowledge additional funding. Corrected acknowledgments appear here.

  • 26 July 2018

    The authors would like to acknowledge additional funding. Corrected acknowledgments appear here.

  • 26 July 2018

    The authors would like to acknowledge additional funding. Corrected acknowledgments appear here.

  • 26 July 2018

    The authors would like to acknowledge additional funding. Corrected acknowledgments appear here.

  • 26 July 2018

    The authors would like to acknowledge additional funding. Corrected acknowledgments appear here.

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Acknowledgements

The authors would like to thank Professor Xiuxiong Chen, Simon Donaldson, Weiyong He, Claude LeBrun and Song Sun for insightful discussions. Haozhao Li and Kai Zheng would like to express their deepest gratitude to Professor Weiyue Ding for his support, guidance and encouragement during the project. Part of this work was done while Haozhao Li was visiting MIT and he wishes to thank MIT for their generous hospitality. H. Li: Supported by NSFC Grant No. 11671370. B. Wang: Supported by NSF Grant DMS-1312836. K. Zheng: Supported by the EPSRC on a Programme Grant entitled “Singularities of Geometric Partial Differential Equations” Reference Number EP/K00865X/1.

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Authors and Affiliations

  1. Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, 230026, Anhui, China

    Haozhao Li

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Bing Wang

  3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Kai Zheng

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  1. Haozhao Li
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  2. Bing Wang
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  3. Kai Zheng
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Correspondence to Bing Wang.

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In memory of Professor Weiyue Ding.

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Li, H., Wang, B. & Zheng, K. Regularity Scales and Convergence of the Calabi Flow. J Geom Anal 28, 2050–2101 (2018). https://doi.org/10.1007/s12220-017-9896-y

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  • DOI: https://doi.org/10.1007/s12220-017-9896-y

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