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

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

2008-11-06

arxiv.org/abs/0811.0991v2

MRL, Vol. 16, no.6 (2009), 935-939.

Mathematics

Differential Geometry

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

Scientific paper

We show that for any solution to the K\"ahler-Ricci flow with positive bisectional curvature on a compact K\"ahler manifold $M^n$, the bisectional curvature has a uniform positive lower bound. As a consequence, the solution converges exponentially fast to an K\"ahler-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.

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