Some curvature estimates of Kähler-Ricci flow

Abstract:

In this work, first we will obtain some local curvature estimates for Kähler-Ricci flow on Kähler manifolds with initial metrics of nonnegative bisectional curvature. As a corollary, we prove that if $g(t)$ is a complete solution of the Kähler Ricci flow which satisfies $|\text {Rm}(g(t))|\leq at^{-\theta }$ for some $0<\theta <2$, $a>0$ and $g(0)$ has nonnegative bisectional curvature, then $g(t)$ also has nonnegative bisectional curvature. This generalizes results in [Amer. J. Math. 140 (2018), pp. 189–220] and [J. Differential Geom. 45 (1997), pp. 94–220]. Using the local curvature estimate, we prove that for a complete solution $g(t)$ of the Kähler-Ricci flow with $g(0)$ to have nonnegative bisectional curvature, to be noncollapsing, and $\sup _{M\times [\tau ,T]} |\text {Rm}(g(t))|<+\infty$ for all $\tau >0$, then the curvature of $g(t)$ is in fact bounded by $at^{-1}$ for some $a>0$. In particular, $g(t)$ has nonnegative bisectional curvature for $t>0$. This result is similar to a result by Simon and Topping in the Kähler category.