Ricci flow under local almost non-negative curvature conditions

TL;DR

This paper establishes local solutions to the Ricci flow under various almost non-negative curvature conditions, extending previous results to higher dimensions and applying the flow to smooth limit spaces.

Contribution

It introduces new local Ricci flow solutions under negative curvature bounds for conditions like 2-non-negative and weakly PIC1, advancing the understanding of curvature preservation.

Findings

Flow exists for a uniform time with bounded negative curvature

Extends 3D results to higher dimensions for 2-non-negative curvature

Shows limit spaces are bi-Hölder homeomorphic to smooth manifolds

Abstract

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. 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 $\textnormal{PIC}_1$ 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-H$\ddot{\textnormal{o}}$lder homeomorphic to a smooth manifold.