Abstract

Let $(M^n, g)$ be a compact $n$ -dimensional ( $n\geq 2$ ) manifold with nonnegative Ricci curvature, and if $n\geq 3,$ then 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 first proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimate for $n= 3$ under ${\rm Rc}\geq 0$ assumption among others. Combining these results, we proved the short-time existence of the Ricci flow on a large class of three-dimensional open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.