A local singularity analysis for the Ricci flow and its applications to Ricci flows with bounded scalar curvature

Reto Buzano; Gianmichele Di Matteo

doi:10.1007/s00526-021-02172-6

A local singularity analysis for the Ricci flow and its applications to Ricci flows with bounded scalar curvature

Calc Var Partial Differ Equ. 2022;61(2):65. doi: 10.1007/s00526-021-02172-6. Epub 2022 Feb 7.

Authors

Reto Buzano^{1

2}, Gianmichele Di Matteo¹

Affiliations

¹ School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS UK.
² Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy.

Abstract

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.

Keywords: 53E20.