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Comptes Rendus Mathématique
Volume 337, n° 12
pages 781-784 (décembre 2003)
Doi : 10.1016/j.crma.2003.09.030
Received : 21 February 2003 ;  accepted : 15 September 2003
Ricci flow on compact Kähler manifolds of positive bisectional curvature

Huai-Dong  Cao ab * ,  Bing-Long  Chen cd ,  Xi-Ping  Zhu cd
aDepartment of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA 
bInstitute for Pure and Applied Mathematics at UCLA, IPAM Building, 460 Portola Plaza, Box 957121, Los Angeles, CA 90095-7121, USA 
cDepartment of Mathematics, Zhongshang University, Guangzhou, 510275, PR China 
dThe Institute of Mathematical Sciences, Unit 601, 6/F, Academic Building No. 1, The Chinese University of Hong Kong, Shatin, Hong Kong 

*Corresponding author.

This Note announces a new proof of the uniform estimate on the curvature of metric solutions to the Ricci flow on a compact Kähler manifold with positive bisectional curvature. This proof does not pre-suppose the existence of a Kähler-Einstein metric on the manifold, unlike the recent work of XiuXiong Chen and Gang Tian. It is based on the Harnack inequality for the Ricci-Kähler flow (see Invent. Math. 10 (1992) 247-263), and also on an estimation of the injectivity radius for the Ricci flow, obtained recently by Perelman. To cite this article: H.-D. Cao et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).


Cette Note annonce une nouvelle démonstration de l'estimée uniforme de la courbure des métriques solutions du flot de Ricci sur une variété kählérienne compacte à courbure bisectionnelle positive. La démonstration proposée ne suppose pas l'existence d'une métrique d'Einstein-Kähler sur la variété, contrairement à un travail récent de XiuXiong Chen et de Gang Tian. Elle s'appuie sur l'inégalité de Harnack pour le flot de Ricci-Kähler (voir Invent. Math. 10 (1992) 247-263), et aussi sur une estimation du rayon d'injectivité du flot de Ricci obtenue récemment par Perelman. Pour citer cet article : H.-D. Cao et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

© 2003  Académie des sciences@@#104156@@

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