On Ricci flows with closed and smooth tangent flows

Abstract

In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler (Structure theory of non-collapsed limits of Ricci flows, 2020. arXiv:2009.03243). The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a question proposed in Chow et al. (The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical surveys and monographs, vol 163. AMS, Providence, 2010), (3) a dichotomy theorem that characterizes ancient Ricci flows admitting a closed and smooth backward sequential limit.