Kähler-Ricci flow with unbounded curvature

Abstract: Let $g(t)$ be a smooth complete solution to the Ricci flow on a noncompact manifold such that $g(0)$ is K\"ahler. We prove that if $|{\rm Rm}(g(t))|_{g(t)}$ is bounded by $a/t$ for some $a>0$, then $g(t)$ is K\"ahler 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 smooth complete solution to the K\"ahler-Ricci flow on a noncompact $n$-dimensional complex manifold such that $g(0)$ has nonnegative holomorphic bisectional curvature and $|{\rm Rm}(g(t))|_{g(t)}\le a(n)/t$, then $g(t)$ has nonnegative holomorphic bisectional curvature for $t>0$. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact K\"ahler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic to ${\Bbb C}^n$; and (ii) there is $\epsilon(n)>0$ depending only on $n$ such that if $(M^n,g_0)$ is a complete noncompact K\"ahler 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 ${\Bbb C}^n$.