Lower bound of Ricci flow's existence time

Abstract

Let $(M^n,g)$ be a compact $n$-dim ($n\ge 2$) manifold with nonnegative Ricci curvature, and if $n\ge 3$ we assume that $(M^n,g)\times \mathbb{R}$ has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on $(M^n,g)$ is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was firstly proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimates for $n=3$ under $Rc\ge 0$ assumption among others. Combining these results, we proved the short time existence of the Ricci flow on a large class of $3$-dim open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.

arXiv comment13 pages, to appear on Bulletin of the London Mathematical Society