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Convergence of the Calabi Flow on Toric Varieties and Related Kähler Manifolds

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Abstract

Let X be a toric manifold with a Delzant polytope P. We show that if (X,P) is analytic uniform K-stable and the curvature is uniformly bounded along the Calabi flow, then the modified Calabi flow converges to an extremal metric exponentially fast. By assuming that the curvature is uniformly bounded along the Calabi flow, we prove a conjecture of Donaldson and a conjecture of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman.

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Notes

  1. Chen and He established the weaker regularity theorem in [10].

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Acknowledgements

The author would like to thank Vestislav Apostolov and Gábor Székelyhidi for many stimulating discussions. He is also grateful to the consistent support of Professor Xiuxiong Chen, Pengfei Guan, and Paul Gauduchon. He would like to thank Si Li, Jeff Streets, and Valentino Tosatti for their interest in this work.

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Authors and Affiliations

  1. CMLS, École Polytechnique, Palaiseau, France

    Hongnian Huang

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  1. Hongnian Huang
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Correspondence to Hongnian Huang.

Additional information

Communicated by Bennett Chow.

The research of the author is financially supported by FMJH (Fondation Mathématique Jacques Hadamard).

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Huang, H. Convergence of the Calabi Flow on Toric Varieties and Related Kähler Manifolds. J Geom Anal 25, 1080–1097 (2015). https://doi.org/10.1007/s12220-013-9457-y

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