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Toric surfaces, \(K\)-stability and Calabi flow

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Abstract

Let \(X\) be a toric surface and \(u\) be a normalized symplectic potential on the corresponding polygon \(P\). Suppose that the Riemannian curvature is bounded by a constant \(C_1\) and \( \int _{\partial P} u d \sigma < C_2, \) then there exists a constant \(C_3\) depending only on \(C_1, C_2\) and \(P\) such that the diameter of \(X\) is bounded by \(C_3\). Moreoever, we can show that there is a constant \(M > 0\) depending only on \(C_1, C_2\) and \(P\) such that Donaldson’s \(M\)-condition holds for \(u\). As an application, we show that if \((X,P)\) is (analytic) relative \(K\)-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.

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Notes

  1. In fact, the initial metric needs to be invariant under the translation in the imaginary part of the complex variables.

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Acknowledgments

The author is grateful to the constant support of Professors Xiuxiong Chen, Pengfei Guan, Vestislav Apostolov and Paul Gauduchon. He is also benefited from the discussions of Si Li, Gábor Székelyhidi and Valentino Tosatti. He would like to thank Jeff Streets for his interests in this work. Thanks also go to the referee for many useful suggestions.

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Authors and Affiliations

  1. CMLS, Ecole Polytechnique, Palaiseau, France

    Hongnian Huang

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  1. Hongnian Huang
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Correspondence to Hongnian Huang.

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

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Huang, H. Toric surfaces, \(K\)-stability and Calabi flow. Math. Z. 276, 953–968 (2014). https://doi.org/10.1007/s00209-013-1228-8

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