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

Albert Chau (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Ka-Fai Li (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Luen-Fai Tam (Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong)

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

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Published 18 March 2016