Ricci flow under local almost non-negative curvature conditions

Abstract

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. Under a non-collapsing assumption, the flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly PIC1 cases, of which the results are new. We complete the discussion of the almost preservation problem by Bamler–Cabezas-Rivas–Wilking, and the 2-non-negative case generalizes a result in 3D by Simon–Topping to higher dimensions.As an application, we use the local Ricci flow to smooth a metric space which is the limit of a sequence of manifolds with the almost non-negative curvature conditions, and show that this limit space is bi-Hölder homeomorphic to a smooth manifold.