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arXiv · 2201.02522

Singularity models in the three-dimensional Ricci flow

S. Brendle
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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).

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S. Brendle. 2022-01-07. Singularity models in the three-dimensional Ricci flow. https://arxiv.org/abs/2201.02522

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