Deforming complete Hermitian metrics with unbounded curvature

Abstract

We produce solutions to the K\"ahler-Ricci flow emerging from complete initial metrics $g_0$ which are $C^0$ Hermitian limits of K\"ahler metrics. Of particular interest is when $g_0$ is K\"ahler with unbounded curvature. We provide such solutions for a wide class of $U(n)$-invariant K\"ahler 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 K\"ahler-Ricci flow has a smooth short time solution starting from any smooth complete $U(n)$-invariant K\"abler 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.