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

In this article we study the Kähler–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 Kähler–Ricci flat in a suitable sense, then the Kähler–Ricci flow 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 C0-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 Kähler manifolds with zero Ricci curvature. II, Invent. Math. 106 (1990), 27–60] on the elliptic Monge–Ampère equation.