The Rate of $${\mathbb {F}}$$-convergence for Ricci Flows with Closed and Smooth Tangent Flows

Abstract

This article is a continuation of Chan et al. ( http://arxiv.org/abs/2109.14763 , 2021), 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 at 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.