Short-time Existence of the Ricci Flow on Noncompact Riemannian Manifolds

In this paper, using the local Ricci flow, we prove the short-time existence of the Ricci flow on noncompact manifolds, whose Ricci curvature has global lower bound and sectional curvature has only local average integral bound. The short-time existence of the Ricci flow on noncompact manifolds was studied by Wan-Xiong Shi in 1990s, who required a point-wise bound of curvature tensors. As a corollary of our main theorem, we get the short-time existence part of Shi’s theorem in this more general context. We get C0 curvature estimates for the local Ricci flow by using so called “local local” curvature estimates and Moser iteration. And in an appendix we give the first detailed proof of the short-time existence of the local Ricci flow.