A C0-estimate for the parabolic Monge-Ampere equation on complete non-compact Kahler manifolds

摘要

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