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

Abstract

In this article we study the Khler-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 Khler-Ricci flat in a suitable sense, then the Khler-Ricci flow has a long time smooth solution g(t) converging smoothly uniformly on compact sets to a complete Khler-Ricci flat metric on M. The main step is to obtain a uniform C 0-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 Khler manifolds with zero Ricci curvature. II, Invent. Math. 106 (1990), 27-60] on the elliptic Monge-Ampère equation. © 2009 Foundation Compositio Mathematica.