## Abstract

In this worK, first we will obtain some local curvature estimates for Kähler-Ricci flow on Kähler manifolds with initial metrics of nonnegative bisectional curvature. As a corollary, we prove that if g(t) is a complete solution of the Kähler Ricci flow which satisfies |Rm(g(t))| ≤ at^{−θ} for some 0 < θ < 2, a > 0 and g(0) has nonnegative bisectional curvature, then g(t) also has nonnegative bisectional curvature. This generalizes results in [Amer. J. Math. 140 (2018), pp. 189–220] and [J. Differential Geom. 45 (1997), pp. 94–220]. Using the local curvature estimate, we prove that for a complete solution g(t) of the Kähler-Ricci flow with g(0) to have nonnegative bisectional curvature, to be noncollapsing, and sup_{M×[τ,T]} |Rm(g(t))| < +∞ for all τ > 0, then the curvature of g(t) is in fact bounded by at^{−1} for some a > 0. In particular, g(t) has nonnegative bisectional curvature for t > 0. This result is similar to a result by Simon and Topping in the Kähler category.

Original language | English (US) |
---|---|

Pages (from-to) | 2641-2654 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 6 |

DOIs | |

State | Published - 2019 |

## Keywords

- Holomorphic bisectional curvature
- Kähler manifold
- Kähler-Ricci flow
- Uniformization

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics