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

Abstract

We show that for any solution gij̄ (t) to the Kähler-Ricci flow with positive bisectional curvature R iījj̄ (t) > 0 on a compact Kähler manifold M n, the bisectional curvature has a uniform positive lower bound Riījj̄ (t) > C > 0. As a consequence, g ij̄ (t) converges exponentially fast in C∞to a Kähler-Einstein metric with positive bisectional curvature as t → ∞, provided we assume that the Futaki-invariant of Mn is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove [22] in which they assumed the stronger condition that the Mabuchi K-energy is bounded from below. © International Press 2009.