Comptes Rendus
Differential Geometry
Uniqueness of extremal Kähler metrics
[Unicité de métriques kählériennes extrémales]
Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 287-290.

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 C1,1 de la solution obtenue antérieurement en montrant qu'elle est C 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 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.11.028

Xiuxiong Chen 1 ; Gang Tian 1

1 Department of Mathematics, University of Wisconsin, Madison, WI 53706-1, USA
@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},
}
TY  - JOUR
AU  - Xiuxiong Chen
AU  - Gang Tian
TI  - Uniqueness of extremal Kähler metrics
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 287
EP  - 290
VL  - 340
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2004.11.028
LA  - en
ID  - CRMATH_2005__340_4_287_0
ER  - 
%0 Journal Article
%A Xiuxiong Chen
%A Gang Tian
%T Uniqueness of extremal Kähler metrics
%J Comptes Rendus. Mathématique
%D 2005
%P 287-290
%V 340
%N 4
%I Elsevier
%R 10.1016/j.crma.2004.11.028
%G en
%F CRMATH_2005__340_4_287_0
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/

[1] S. Bando; T. Mabuchi Uniqueness of Kähler Einstein metrics modulo connected group actions, Algebraic Geometry, Adv. Stud. Pure Math., 1987, pp. 11-40

[2] E.D. Bedford; T.A. Taylor The Drichelet problem for the complex Monge–Ampere operator, Invent. Math., Volume 37 (1976), pp. 1-44

[3] E. Calabi Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton University Press, 1982, pp. 259-290

[4] X.X. Chen Space of Kähler metrics, J. Differential Geom., Volume 56 (2000), pp. 189-234

[5] X.X. Chen; G. Tian Partial regularity of homogeneous complex Monge–Ampere equations, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005) no. 5

[6] X.X. Chen, G. Tian, Geometry of Kähler metrics and foliations by holomorphic discs, Preprint, 2004

[7] S.K. Donaldson Holomorphic discs and the complex Monge–Ampere equation, J. Symplectic Geometry, Volume 1 (2002), pp. 171-196

[8] S.K. Donaldson Scalar curvature and projective embeddings. I, J. Differential Geom., Volume 59 (2001), pp. 479-522

[9] S.K. Donaldson Scalar curvature and projective embeddings. II | arXiv

[10] T. Mabuchi Some Symplectic geometry on compact Kähler manifolds I, Osaka J. Math., Volume 24 (1987), pp. 227-252

[11] T. Mabuchi Stability of extremal Kähler manifolds | arXiv

[12] S. Paul, G. Tian, Algebraic and analytic K-stability, Preprint, 2004

[13] S. Semmes Complex Monge–Ampere and sympletic manifolds, Amer. J. Math., Volume 114 (1992), pp. 495-550

[14] G. Tian Kähler–Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997), pp. 1-39

[15] G. Tian Bott–Chern forms and geometric stability, Discrete Contin. Dynam. Systems, Volume 6 (2000), pp. 1-39

[16] G. Tian; X.H. Zhu Uniqueness of Kähler–Ricci solitons, Acta Math., Volume 184 (2000), pp. 271-305

Cité par Sources :

Commentaires - Politique