TY - JOUR

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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U2 - 10.1016/j.crma.2004.11.028

DO - 10.1016/j.crma.2004.11.028

M3 - Article

AN - SCOPUS:13844310726

VL - 340

SP - 287

EP - 290

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 4

ER -