A C⁰-estimate for the parabolic Monge-Amp\`{e}re equation on complete non-compact K\"ahler manifolds

In this article we study the K\"ahler Ricci flow, the corresponding parabolic Monge Amp\`{e}re equation and complete non-compact K\"ahler Ricci flat manifolds. In our main result Theorem \ref{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 \eqref{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 $C^0$-estimates for the corresponding parabolic Monge Amp\`{e}re equation. Our results on this can be viewed as a parabolic version of the main results in \cite{TY3} on the elliptic Monge Amp\`{e}re equation.