A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature
Published 2020-06-29Version 1
We prove that Ricci flows with bounded scalar curvature cannot develop finite time singularities in dimensions less than eight. In order to study such flows in higher dimensions, we then 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. Combining these and other results, we then show that for Ricci flows with bounded scalar curvature in higher dimensions the singular set has codimension eight in a suitable sense.