  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.
