Calculus of Variations and Geometric Measure Theory

R. Buzano - G. Di Matteo

A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature

created by dimatteo on 01 Sep 2020
modified by muller on 13 Jun 2022


Published Paper

Inserted: 1 sep 2020
Last Updated: 13 jun 2022

Journal: Calculus of Variations and Partial Differential Equations
Volume: 61
Year: 2022
Doi: 10.1007/s00526-021-02172-6

ArXiv: 2006.16227 PDF


We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum's result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Finally, we show some applications of the theory to Ricci flows with bounded scalar curvature.