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Zbl 1097.53045
Chen, Xiuxiong; Tian, Gang
Ricci flow on Kähler-Einstein manifolds. (English)
[J] Duke Math. J. 131, No. 1, 17-73 (2006). ISSN 0012-7094

There is a long-standing problem in Ricci flow: on a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has positive bisectional curvature? In this article authors give a complete affirmative answer to this problem. Theorem 1.1: Let $M$ be a Kähler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive bisectional curvature at least at one point, then the Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric with constant bisectional curvature. This problem was completely solved by R. S. Hamilton in the case of Riemann surfaces. As a direct consequence, the authors have the following Corollary 1.3: The space of Kähler metrics with nonnegative bisectional curvature (and positive at least at one point) is path connected. The space of metrics with a nonnegative curvature operator (and positive at least at one point) is also path connected. Theorem 1.4. Let $M$ be any Kähler-Einstein orbifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive bisectional curvature at least at one point, then the Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric with constant bisectional curvature. Moreover, $M$ is a global quotient of $\Bbb CP^{n}.$
[V. V. Chueshev (Kemerovo)]

MSC 2000:: *53C44 Geometric evolution equations (mean curvature flow)
32Q20 Kähler-Einstein manifolds
53C25 Special Riemannian manifolds
53C55 Complex differential geometry (global)

Keywords: compact Kähler-Einstein manifold; the Kähler-Ricci flow; Kähler-Einstein metric with nonnegative bisectional curvature

Cited in: Zbl 1237.53066 Zbl 1215.53062

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