Acta Mathematica

Volume 225 (2020)

Number 1

Ancient solutions to the Ricci flow in dimension $3$

Pages: 1 – 102



Simon Brendle (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)


It follows from work of Perelman that any finite-time singularity of the Ricci flow on a compact $3$-manifold is modeled on an ancient $\varkappa$-solution.

We prove that every non-compact ancient $\varkappa$-solution in dimension $3$ is isometric to a family of shrinking cylinders (or a quotient thereof), or to the Bryant soliton. This confirms a conjecture of Perelman.

Received 31 January 2019

Accepted 1 March 2020

Published 4 November 2020