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The Dirichlet and the weighted metrics for the space of Kähler metrics

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In this work we study the intrinsic geometry of the space of Kähler metrics under various Riemannian metrics and the corresponding variational structures. The first part is on the Dirichlet metric. A motivation for the study of this metric comes from Chen and Zheng (J Reine Angew Math, 674:195–251, 2013); there, Chen and the second author showed that the pseudo-Calabi flow is the gradient flow of the \(K\)-energy when \(\mathcal {H}\) is endowed precisely with the Dirichlet metric. The second part is on the family of weighted metrics, whose distinguished element is the Calabi metric studied in Calamai (Math Ann 353:373–402, 2012). We investigate as well their geometric properties. Then we focus on the constant weight metric. We use it to give an alternative proof of Calabi’s uniqueness of the Kähler–Einstein metrics (cf. Theorem 5.9), when \(C_1\le 0\). We also introduce a new functional called \(G\)-functional and we prove that its gradient flow has long time existence and also converges to the Kähler–Einstein metric when \(C_1\le 0\) (cf. Theorem 5.10).

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Acknowledgments

The authors would like to express their deepest gratitude to Professor Claudio Arezzo and Professor Xiuxiong Chen for helpful discussion on this paper.

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Author notes

  1. Kai Zheng

    Present address: Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni, 67/a, 50134, Florence, Italy

    Simone Calamai

  2. Institut Fourier, Université Joseph Fourier (Grenoble I), UMR 5582 CNRS-UJF, BP 74, 38402, Saint-Martin-d’Hères, France

    Kai Zheng

Authors
  1. Simone Calamai
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  2. Kai Zheng
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Correspondence to Kai Zheng.

Additional information

S. Calamai is supported by GNSAGA of INdAM, and by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”. K. Zheng author is partially supported by ANR project “Flots et Opérateurs Géométriques” (ANR-07-BLAN-0251-01).

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Calamai, S., Zheng, K. The Dirichlet and the weighted metrics for the space of Kähler metrics. Math. Ann. 363, 817–856 (2015). https://doi.org/10.1007/s00208-015-1188-x

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