Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server