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The Kähler-Ricci flow on non-compact manifolds Li, Ka-Fai


We first study the general theory of Kähler-Ricci flow on non-compact complex manifolds. By using a parabolic Schwarz lemma and a local scalar curvature estimate, we prove a general existence theorem for Kähler metrics lying in the C^\infty _\loc closure of complete bounded curvature Kähler metrics that are uniformly equivalent to a fixed background metric. In particular we do not assume any curvature bounds. Next, we compare the maximal existence time of two complete bounded curvature solutions by using the equivalence of the initial metrics and using this, we also estimate the maximal existence time of a complete bounded curvature solution in terms of the curvature bound of a background metric. We also prove a uniqueness theorem for Kähler-Ricci flow which slightly improves the result of [13] in the Kähler case. We apply the above results to study the Kähler-Ricci flow on some specific non-compact complex manifolds. We first study the Kähler-Ricci flow on C^n. By applying our general existence theorem and existence time estimate, we show that any complete non-negatively curved U(n)-invariant Kähler metric admits a longtime U(n)-invariant solution to the Kähler-Ricci flow, and the solution converges to the standard Euclidean metric after rescaling. Then we study the Kähler-Ricci flow on a quasi-projective manifold. By modifying the approximation theorem of [1] and applying a general existence theorem of Lott-Zhang [23], we construct a Kähler-Ricci flow solution starting from certain smooth Kähler metrics. In particular, if the metric is the restriction of a smooth Kähler metric in the ambient space, then the solution instantaneously becomes complete and has cusp singularity at the divisor. We also produce a solution starting from some complete metrics that may not have bounded curvature, and the solution is likewise complete with cusp singularity for positive time. On the other hand, if the initial data has bounded curvature and is asymptotic to the standard cusp model in a certain sense, we find the maximal existence time of the corresponding complete bounded curvature solution to the Kähler-Ricci flow.

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