A $C^0$-estimate for the parabolic Monge-Ampère equation on complete non-compact Kähler manifolds

Abstract

In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete non-compact Kähler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M,g)$ is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler Ricci flow \eqref{KRF} 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 $C^0$-estimates for the corresponding parabolic Monge Ampère equation. Our results on this can be viewed as a parabolic version of the main results in \cite{TY3} on the elliptic Monge Ampère equation.

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• Quasi-asymptotically conical Calabi-Yau manifolds