Ricci flow on compact Kähler manifolds of positive bisectional curvature

Abstract

This Note announces a new proof of the uniform estimate on the curvature of metric solutions to the Ricci flow on a compact Kähler manifold with positive bisectional curvature. This proof does not pre-suppose the existence of a Kähler–Einstein metric on the manifold, unlike the recent work of XiuXiong Chen and Gang Tian. It is based on the Harnack inequality for the Ricci–Kähler flow (see Invent. Math. 10 (1992) 247–263), and also on an estimation of the injectivity radius for the Ricci flow, obtained recently by Perelman.