Deforming complete Hermitian metrics with unbounded curvature

Abstract

We produce solutions to the Kähler-Ricci flow emerging from complete initial metrics $g_0$ which are $C^0$ Hermitian limits of Kähler metrics. Of particular interest is when $g_0$ is Kähler with unbounded curvature. We provide such solutions for a wide class of $U(n)$-invariant Kähler metrics $g_0$ on $n$ dimensional complex Euclidean space, many of which having unbounded curvature. As a special case we have the following Corollary: The Kähler-Ricci flow has a smooth short time solution starting from any smooth complete $U(n)$-invariant Käbler metric on $\C^n$ with either non-negative or non-positive holomorphic bisectional curvature, and the solution exists for all time in the case of non-positive curvature.

32 pages; v2 minor corrections and editions, reference added, Corollary 4.3 removed due to mistake in proof, precise dependence of $T$ on initial constants removed from statement of Theorem 5.4

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• Kähler-Ricci flow with unbounded curvature