Time Zero Regularity of Ricci Flow

Abstract

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 that was given in [26]. We prove a positive result in the case that the flow satisfies a lower ${ {\textrm {K}_{\textrm {IC}_1}}}$ 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.