The rate of $\mathbb{F}$-convergence for Ricci flows with closed and smooth tangent flows

Abstract

This article is a continuation of [CMZ21b], where we proved that a Ricci flow with a closed and smooth tangent flow has unique tangent flow, and its corresponding forward or backward modified Ricci flow converges in the rate of $t^{-β}$ for some $β>0$. In this article, we calculate the corresponding $\mathbb{F}$-convergence rate: after being scaled by a factor $λ>0$, a Ricci flow with closed and smooth tangent flow is $|\log λ|^{-θ}$ close to its tangent flow in the $\mathbb{F}$-sense, where $θ$ is a positive number, $λ\gg 1$ in the blow-up case, and $λ\ll 1$ in the blow-down case.