Time zero regularity of Ricci flow

TL;DR

This paper investigates the conditions under which Ricci flows become smooth immediately after initial time, providing curvature estimates and demonstrating cases where smoothness may fail, with applications to WPIC1 manifolds.

Contribution

It establishes new curvature estimates and conditions ensuring initial regularity of Ricci flows, especially under lower IC1 curvature bounds, and applies results to WPIC1 manifold limits.

Findings

Curvature estimates for Ricci flows with initial data

Failure cases for immediate regularity

WPIC1 limits preserve WPIC1 structure

Abstract

We consider the problem of when a smooth Ricci flow, for positive time, that attains smooth initial data in a weak sense must be smooth down to the initial time. We obtain curvature estimates for an example where this fails. We prove a positive result in the case that the flow satisfies a lower IC1 curvature bound, equivalent to a lower Ricci bound in three dimensions. As an application, we prove that Gromov-Hausdorff limits of WPIC1 manifolds are WPIC1.