The Dirichlet and the weighted metrics for the space of K?hler metrics

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• 作者：Simone Calamai ; Kai Zheng
• 关键词：53C55 ; 53C22
• 刊名：Mathematische Annalen
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：363
• 期：3-4
• 页码：817-856
• 全文大小：658 KB
• 参考文献：1.Arezzo, C., Tian, G.: Infinite geodesic rays in the space of K?hler potentials (English summary)
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  6.Calamai, S.: The Calabi metric for the space of K?hler metrics. Math. Ann. 353, 373-02 (2012)MATH MathSciNet CrossRef
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  13.Chen, X.X., Zheng, K.: The pseudo-Calabi flow. J. Reine Angew. Math. 674, 195-51 (2013)MATH MathSciNet
  14.Donaldson, S.K.: Symmetric spaces, K?hler geometry and Hamiltonian dynamics. In: Northern California Symplectic Geometry Seminar, pp. 13-3. American Mathematical Society Translations (2), vol. 196. American Mathematical Society, Providence (1999)
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• 作者单位：Simone Calamai (1)
  Kai Zheng (2) (3)
  
  1. Dipartimento di Matematica e Informatica “Ulisse Dini- Università degli Studi di Firenze, Viale Morgagni, 67/a, 50134, Florence, Italy
  2. Institut Fourier, Université Joseph Fourier (Grenoble I), UMR 5582 CNRS-UJF, BP 74, 38402, Saint-Martin-d’Hères, France
  3. Institut für Differentialgeometrie, Leibniz Universit?t Hannover, Welfengarten 1, 30167, Hannover, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1807

文摘

In this work we study the intrinsic geometry of the space of K?hler 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, 674:195-51, 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 353:373-02, 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 K?hler–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 K?hler–Einstein metric when \(C_1\le 0\) (cf. Theorem 5.10). Mathematics Subject Classification 53C55 53C22