Singularity Models in the Three-Dimensional Ricci Flow

Abstract

AbstractThe Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. The main goal is to understand singularity formation. In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension 3. More precisely, Perelman showed that every finite-time singularity to the Ricci flow in dimension 3 is modeled on an ancient \(\kappa \)-solution. Moreover, Perelman proved a structure theorem for ancient \(\kappa \)-solutions in dimension 3. In this survey, we discuss recent developments which have led to a complete classification of all the singularity models in dimension 3. Moreover, we give an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3 (originally proved by the author in 2012).KeywordsRicci flowRicci solitonAncient solution