A C0-estimate for the parabolic Monge-Amp\`ere equation on complete non-compact K\"ahler manifolds
Abstract
In this article we study the K\"ahler Ricci flow, the corresponding parabolic Monge Amp\`ere equation and complete non-compact K\"ahler Ricci flat manifolds. In our main result Theorem mainthm we prove that if (M, g) is sufficiently close to being K\"ahler Ricci flat in a suitable sense, then the K\"ahler Ricci flow KRF has a long time smooth solution g(t) converging smoothly uniformly on compact sets to a complete K\"ahler Ricci flat metric on M. The main step is to obtain a uniform C0-estimates for the corresponding parabolic Monge Amp\`ere equation. Our results on this can be viewed as a parabolic version of the main results in TY3 on the elliptic Monge Amp\`ere equation.
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