Differential Geometry
Ricci flow on compact Kähler manifolds of positive bisectional curvature
Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 781-784.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.09.030
Cao, Huai-Dong 1, 2; Chen, Bing-Long 3, 4; Zhu, Xi-Ping 3, 4

1 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
2 Institute for Pure and Applied Mathematics at UCLA, IPAM Building, 460 Portola Plaza, Box 957121, Los Angeles, CA 90095-7121, USA
3 Department of Mathematics, Zhongshang University, Guangzhou, 510275, PR China
4 The Institute of Mathematical Sciences, Unit 601, 6/F, Academic Building No. 1, The Chinese University of Hong Kong, Shatin, Hong Kong
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Cao, Huai-Dong; Chen, Bing-Long; Zhu, Xi-Ping. Ricci flow on compact Kähler manifolds of positive bisectional curvature. Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 781-784. doi : 10.1016/j.crma.2003.09.030. http://www.numdam.org/articles/10.1016/j.crma.2003.09.030/

[1] Bando, S. Compact 3-folds with nonnegative bisectional curvature, J. Differential Geom., Volume 19 (1984), pp. 283-297

[2] Bishop, R.L.; Goldberg, S.I. On the second cohomology group of a Kähler manifold of positive curvature, Proc. Amer. Math. Soc., Volume 16 (1965), pp. 119-122

[3] Cao, H.-D. Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds, Invent. Math., Volume 81 (1985), pp. 359-372

[4] Cao, H.-D. On Hanarck's inequalities for the Kähler–Ricci flow, Invent. Math., Volume 109 (1992), pp. 247-263

[5] Chen, X.; Tian, G. Ricci flow on Kähler–Einstein surfaces, Invent. Math., Volume 147 (2002), pp. 487-544

[6] X. Chen, G. Tian, Ricci flow on Kähler–Einstein manifolds, Preprint

[7] Chow, B. The Ricci flow on the 2-sphere, J. Differential Geom., Volume 33 (1991), pp. 325-334

[8] Hamilton, R.S. The Ricci flow on surfaces, Mathematics and General Relativity, Contemp. Math., 71, 1988, pp. 237-261

[9] Hamilton, R.S. An isoperimetric estimate for the Ricci flow on the 2-sphere, Modern Methods in Complex Analysis (Princeton, NJ, 1992), Ann. of Math. Stud., 137, Princeton Univ. Press, 1995, pp. 201-222

[10] Mok, N. The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom., Volume 27 (1988), pp. 179-214

[11] Mori, S. Projective manifolds with ample tangent bundles, Ann. Math., Volume 76 (1979), pp. 213-234

[12] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Preprint

[13] Schoen, R.; Yau, S.-T. Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994

[14] Siu, Y.-T.; Yau, S.-T. Compact Kähler manifolds of positive bisectional curvature, Inven. Math., Volume 59 (1980), pp. 189-204

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

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