Ricci flow on Kähler-Einstein manifolds

Details Ricci flow on Kähler-Einstein manifolds Ricci flow on Kähler-Einstein manifolds

2001-08-27

arxiv.org/abs/math/0108179v2

Mathematics

Differential Geometry

49 pages. This is a revised version. Sections 4 and 5 are simplified and streamlined

Scientific paper

10.1007/s002220100181

In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the K\"ahler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functionals in case of Kaehler manifolds. On K\"ahler-Einstein manifold with positive scalar curvature, if the initial metric has positive bisectional curvature, we prove that these functionals have a uniform lower bound, via the effective use of Tian's inequality. Consequently, we prove the following theorem: Let $M$ be a K\"ahler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler Ricci flow will converge exponentially fast to a K\"ahler-Einstein metric with constant bisectional curvature. Such a result holds for K\"ahler-Einstein orbifolds.

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