Lower bound of Ricci flow's existence time

TL;DR

This paper establishes a lower bound for the existence time of Ricci flow on certain compact manifolds with nonnegative Ricci curvature, providing new proofs and curvature estimates, and demonstrating short-time existence on specific open 3-manifolds.

Contribution

It offers an alternative proof for the lower bound of Ricci flow existence time and derives interior curvature estimates, extending short-time existence results to a broad class of open 3-manifolds.

Findings

Lower bound for Ricci flow existence time on manifolds with nonnegative Ricci curvature.

Interior curvature estimates for 3-manifolds with nonnegative Ricci curvature.

Short-time existence of Ricci flow on certain open 3-manifolds with exhaustion coverings.

Abstract

Let $(M^n, g)$ be a compact $n$-dim ($n\geq 2$) manifold with nonnegative Ricci curvature, and if $n\geq 3$ 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 $Rc\geq 0$ 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.