Short-time existence of the Ricci flow on noncompact Riemannian manifolds

Abstract

In this paper, we give the first detailed proof of the short-time existence of Deane Yang’s local Ricci flow. Then, 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 with bounded curvature was studied by Wan-Xiong Shi in the 1990s. As a corollary of our main theorem, we obtain the short-time existence part of Shi’s theorem in this more general context.