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A C0-estimate for the parabolic Monge–Ampère equation on complete non-compact Kähler manifolds

Part of: Global differential geometry Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  11 December 2009

Albert Chau  and
Luen-Fai Tam
Albert Chau
Affiliation:
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2 (email: [email protected])
Luen-Fai Tam
Affiliation:
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China (email: [email protected])
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Abstract

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In this article we study the Kähler–Ricci flow, the corresponding parabolic Monge–Ampère equation and complete non-compact Kähler–Ricci flat manifolds. Our main result states that if (M,g) is sufficiently close to being Kähler–Ricci flat in a suitable sense, then the Kähler–Ricci flow has a long time smooth solution g(t) converging smoothly uniformly on compact sets to a complete Kähler–Ricci flat metric on M. The main step is to obtain a uniform C0-estimate for the corresponding parabolic Monge–Ampère equation. Our results on this can be viewed as parabolic versions of the main results of Tian and Yau [Complete Kähler manifolds with zero Ricci curvature. II, Invent. Math. 106 (1990), 27–60] on the elliptic Monge–Ampère equation.

Keywords

non-compact Kähler–Einstein metricsKähler–Ricci flowparabolic Monge–Ampère equation

MSC classification

Secondary: 53C55: Hermitian and Kählerian manifolds 58J35: Heat and other parabolic equation methods
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 259 - 270
Copyright
Copyright © Foundation Compositio Mathematica 2009

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