Uniqueness of extremal Kähler metrics

Xiuxiong Chen; Gang Tian

doi:10.1016/j.crma.2004.11.028

Uniqueness of extremal Kähler metrics

Xiuxiong Chen
, Gang Tian

Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	287-290
Number of pages	4
Journal	Comptes Rendus Mathematique
Volume	340
Issue number	4
DOIs	https://doi.org/10.1016/j.crma.2004.11.028
State	Published - Feb 15 2005

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1016/j.crma.2004.11.028

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title = "Uniqueness of extremal K{\"a}hler metrics",

abstract = "In the infinite dimensional space of K{\"a}hler potentials, the geodesic equation of disc type is a complex homogenous Monge-Amp{\`e}re 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 C1,1 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 K{\"a}hler metrics and to establish a lower bound for the modified K-energy if the underlying K{\"a}hler class admits an extremal K{\"a}hler metric.",

author = "Xiuxiong Chen and Gang Tian",

note = "Funding Information: 1 Both authors are supported by NSF research grants and the second author is also supported partially by a Simons fund grant.",

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T1 - Uniqueness of extremal Kähler metrics

AU - Chen, Xiuxiong

AU - Tian, Gang

N1 - Funding Information: 1 Both authors are supported by NSF research grants and the second author is also supported partially by a Simons fund grant.

PY - 2005/2/15

Y1 - 2005/2/15

N2 - 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 C1,1 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.

AB - 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 C1,1 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.

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DO - 10.1016/j.crma.2004.11.028

M3 - Article

AN - SCOPUS:13844310726

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EP - 290

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

IS - 4

ER -

Uniqueness of extremal Kähler metrics

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