*(English)*Zbl 1097.53045

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 $\u2102{P}^{n}\xb7$

##### MSC:

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) |

32Q20 | Kähler-Einstein manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C55 | Hermitian and Kählerian manifolds (global differential geometry) |