Ricci flow on Kähler manifolds (Q2712612)

The Kähler-Ricci flow on a Kähler manifold always has a global solution and such a solution converges to a Kähler-Einstein metric if the first Chern class of the underlying Kähler manifold is zero or negative [\textit{H.-D. Cao}, Invent. Math. 81, 359-372 (1985; Zbl 0574.53042)]. Nevertheless, if the first Chern class is positive, the solution of a Kähler-Ricci flow may not converge to any Kähler-Einstein metric. A natural and interesting problem is whether or not the Kähler-Ricci flow on a compact Kähler-Einstein manifold converges to a Kähler-Einstein metric.NEWLINENEWLINENEWLINEIn this note, the authors announce the following result: Let \(M\) be a Kähler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and the curvature is positive at least at one point, then the Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric with constant bisectional curvature.