Uniqueness of compact ancient solutions to three-dimensional Ricci flow

Abstract

In this paper, we study the classification of $\kappa$-noncollapsed ancient solutions to three-dimensional Ricci flow on $S^3$. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.

This paper supersedes the paper arXiv:1904.07835 by Brendle and the paper arXiv:1907.01928 by Daskalopoulos and Sesum. We also remove the assumption on reflection symmetry that was present in the latter paper

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• Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow
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• Classification of bubble-sheet ovals in ${\mathbb{R}}^4$
• Ancient solutions to the Kähler Ricci flow
• Ancient solutions to the Ricci flow with isotropic curvature conditions
• Ancient finite entropy flows by powers of curvature in ${\mathbb{R}}^2$
• Hearing the shape of ancient noncollapsed flows in ${\mathbb{R}}^4$