An existence time estimate for K\"ahler-Ricci flow

Abstract

Fix a complete noncompact \K manifold $(M^n,h_0)$ with bounded curvature. Let $g(t)$ be a bounded curvature solution to the \KR flow starting from some $g_0$ uniformly equivalent to $h_0$. We estimate the existence time of $g(t)$ together with $C^0$ bounds and curvature bounds, where the estimates depend only on $h_0$ and the $C^0$ distance between $g_0$ and $h_0$. We also generalize these results to cases when $g_0$ may have unbounded curvature.