Singularity models in the three-dimensional Ricci flow

TL;DR

This paper surveys recent advances in classifying singularity models in three-dimensional Ricci flow, building on Perelman's foundational work and providing new proofs for known classifications.

Contribution

It offers a complete classification of all 3D Ricci flow singularity models and presents an alternative proof for steady gradient Ricci solitons.

Findings

Complete classification of 3D singularity models

Alternative proof for steady gradient Ricci solitons

Enhanced understanding of singularity formation in Ricci flow

Abstract

The 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).