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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 2

### Deforming complete Hermitian metrics with unbounded curvature

Pages: 267 – 292

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n2.a3

#### Authors

#### 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 $\mathbb{C}^n$, 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 Kähler metric on $\mathbb{C}^n$ with either non-negative or non-positive holomorphic bisectional curvature, and the solution exists for all time in the case of non-positive curvature.

#### Keywords

Kähler–Ricci flow, parabolic Monge–Ampère equation, $U(n)$ invariant Kähler metrics

#### 2010 Mathematics Subject Classification

53C55, 58J35