[Unicité de métriques kählériennes extrémales]
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 de la solution obtenue antérieurement en montrant qu'elle est 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 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.
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@article{CRMATH_2005__340_4_287_0,
author = {Xiuxiong Chen and Gang Tian},
title = {Uniqueness of extremal {K\"ahler} metrics},
journal = {Comptes Rendus. Math\'ematique},
pages = {287--290},
publisher = {Elsevier},
volume = {340},
number = {4},
year = {2005},
doi = {10.1016/j.crma.2004.11.028},
language = {en},
}
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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