A note on compact K\"ahler-Ricci flow with positive bisectional curvature

TL;DR

This paper proves that solutions to the K"ahler-Ricci flow with positive bisectional curvature on compact K"ahler manifolds have a uniform positive lower bound on curvature and converge exponentially to a K"ahler-Einstein metric under certain conditions.

Contribution

It establishes a uniform positive lower bound on bisectional curvature and exponential convergence to K"ahler-Einstein metrics assuming zero Futaki invariant, improving previous results.

Findings

Bisectional curvature has a uniform positive lower bound.

Solutions converge exponentially to K"ahler-Einstein metrics.

Convergence holds under zero Futaki invariant condition.

Abstract

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.