Ricci flow on Kähler-Einstein surfaces

Abstract.

In this paper, we prove that if M is a Kähler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kähler-Ricci flow converges to a Kähler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kähler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kähler Ricci flow: On a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kähler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.