A note on compact Kähler-Ricci flow with positive bisectional curvature

Abstract

We show that for any solution to the Kähler-Ricci flow with positive bisectional curvature on a compact Kähler manifold $M^n$, the bisectional curvature has a uniform positive lower bound. As a consequence, the solution converges exponentially fast to an Kähler-Einstein metric with positive bisectional curvature as t tends to the infinity, provided we assume the Futaki-invariant of $M^n$ is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove in which they assumed the stronger condition that Mabuchi K-energy is bounded from below.

Simplified the proof of Theorem 1.1; added references and corrected some typos

References in corpus (3)

• The Kähler-Ricci flow and the $\bar{\partial }$ operator on vector fields
• A note on Kähler-Ricci soliton
• A Simple Proof for the Generalized Frankel Conjecture