Acta Mathematica

Volume 228 (2022)

Number 1

Uniqueness and stability of Ricci flow through singularities

Pages: 1 – 215



Richard H. Bamler (Department of Mathematics, University of California, Berkeley, Cal., U.S.A.)

Bruce Kleiner (Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)


We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3‑manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman’s conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of 3‑manifolds—in particular to the generalized Smale conjecture—which will appear in a subsequent paper.

The first author was supported by a Sloan Research Fellowship and NSF grant DMS-1611906. The research was in part conducted while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester. The second author was supported by NSF grants DMS-1405899 and DMS-1406394, and a Simons Collaboration grant.

Received 10 April 2019

Accepted 4 January 2022

Published 1 July 2022