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

TL;DR

This paper determines the rate at which Ricci flows with closed, smooth tangent flows converge in the -sense, showing a logarithmic decay rate after appropriate scaling, extending previous convergence results.

Contribution

It computes the -convergence rate for Ricci flows with smooth tangent flows, revealing a logarithmic decay in the scaled flows, building on prior convergence rate results.

Findings

-convergence rate is proportional to | log  mbda|^{- heta}.

Convergence rate applies in both blow-up and blow-down scenarios.

Provides explicit decay rate in -sense for Ricci flows with 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 $t^{-\beta}$ for some $\beta>0$. In this article, we calculate the corresponding $\mathbb{F}$-convergence rate: after being scaled by a factor $\lambda>0$, a Ricci flow with closed and smooth tangent flow is $|\log \lambda|^{-\theta}$ close to its tangent flow in the $\mathbb{F}$-sense, where $\theta$ is a positive number, $\lambda\gg 1$ in the blow-up case, and $\lambda\ll 1$ in the blow-down case.