Ancient solutions to Kähler-Ricci flow

In this paper, we prove that any non-flat ancient solution to Kähler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also classify all complete gradient shrinking solitons with nonnegative bisectional curvature. Both results generalize the corresponding earlier results of Perelman in [P1] and [P2]. The results then are applied to study the geometry and function theory of complete Kähler manifolds with nonnegative bisectional curvature via Kähler-Ricci flow. A compactness result on ancient solutions to Kähler-Ricci flow is also obtained.

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Published 1 January 2005