Ricci flow on compact K\"ahler manifolds of positive bisectional curvature

Abstract

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact K\"ahler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of the uniform estimate does not rely on the exsitence of K\"ahler-Einstein metrics on $M^n$, but instead on the first author's Harnack inequality for the K\"ahler-Ricc flow, and a very recent local injectivity radius estimate of Perelman for the Ricci flow.