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
SN - 1631-073X
VL - 340
SP - 287
EP - 290
JO - Comptes Rendus Mathematique
JF - Comptes Rendus Mathematique
IS - 4
ER -