  Let $g(t)$ be a complete solution to the Ricci flow on a noncompact manifold
such that $g(0)$ is Kahler. We prove that if $|Rm(g(t))|_{g(t)}\le a/t$ for
some $a>0$, then $g(t)$ is Kahler for $t>0$. We prove that there is a constant
$ a(n)>0$ depending only on $n$ such that the following is true: Suppose $g(t)$
is a complete solution to the Kahler-Ricci flow on a noncompact $n$-dimensional
complex manifold such that $g(0)$ has nonnegative holomorphic bisectional
curvature and such that $|Rm(g(t))|_{g(t)}\le a(n)/t$, then $g(t)$ has
nonnegative holomorphic bisectional curvature for $t>0$. These generalize the
results by W. S. Shi. As corollaries, we prove that (i) any complete noncompact
Kahler manifold with nonnegative complex sectional curvature with maximum
volume growth is biholomorphic to $ C^n$; and (ii) there is $\epsilon(n)>0$
depending only on $n$ such that if $(M^n,g_0)$ is a complete noncompact Kahler
manifold of complex dimension $n$ with nonnegative holomorphic bisectional
curvature and maximum volume growth and if $(1+\epsilon(n))^{-1}h\le g_0\le
(1+\epsilon(n))h$ for some Riemannian metric $h$ with bounded curvature, then
$M$ is biholomorphic to $C^n$.

Differential Geometry