Uniqueness of extremal Kähler metrics

Xiuxiong Chen; Gang Tian

doi:10.1016/j.crma.2004.11.028

Differential Geometry

Uniqueness of extremal Kähler metrics
[Unicité de métriques kählériennes extrémales]

Présenté par : Jean-Michel Bismut

Xiuxiong Chen ¹ ; Gang Tian ¹

¹ Department of Mathematics, University of Wisconsin, Madison, WI 53706-1, USA

Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 287-290.

Résumé
Abstract

Dans l'espace de dimension infinie des potentiels de Kähler, l'équation géodésique de type disque est une équation de Monge–Ampère complexe homogène. Le résulat de régularité partielle établi dans cette note permet de renforcer le caractère $C^{1, 1}$ de la solution obtenue antérieurement en montrant qu'elle est $C^{\infty}$ presque partout. On démontre que la K-énergie est sous-harmonique sur une telle solution. On utilise ce résultat pour montrer l'unicité de la métrique de Kähler extrémale et pour établir une borne inférieure pour la K-énergie, quand la classe de Kähler sous-jacente admet une métrique Kählérienne extrémale.

In the infinite dimensional space of Kähler potentials, the geodesic equation of disc type is a complex homogenous Monge–Ampère equation. The partial regularity theory established by Chen and Tian [C. R. Acad. Sci. Paris, Ser. I 340 (5) (2005)] amounts to an improvement of the regularity of the known $C^{1, 1}$ solution to the geodesic of disc type to almost everywhere smooth. For such an almost smooth solution, we prove that the K-energy functional is sub-harmonic along such a solution. We use this to prove the uniqueness of extremal Kähler metrics and to establish a lower bound for the modified K-energy if the underlying Kähler class admits an extremal Kähler metric.

Reçu le : 2004-11-04
Accepté le : 2004-11-05
Publié le : 2005-01-22

DOI : 10.1016/j.crma.2004.11.028

Affiliations des auteurs :

Xiuxiong Chen ¹ ; Gang Tian ¹

¹ Department of Mathematics, University of Wisconsin, Madison, WI 53706-1, USA

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Xiuxiong Chen; Gang Tian. Uniqueness of extremal Kähler metrics. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 287-290. doi : 10.1016/j.crma.2004.11.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.028/

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