Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

Brendle, Simon; Naff, Keaton

doi:10.2140/gt.2023.27.153

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Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

Simon Brendle and Keaton Naff

Geometry & Topology 27 (2023) 153–226

DOI: 10.2140/gt.2023.27.153

Abstract

We extend work of the first author on the uniqueness of ancient $κ$ –solutions to higher dimensions. In dimensions $n \geq 4$ , an ancient $κ$ –solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2, has bounded curvature and is $κ$ –noncollapsed. We show that the only noncompact ancient $κ$ –solutions up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.

Keywords

Ricci flow, rotational symmetry, ancient solutions, higher dimensions

Mathematical Subject Classification

Primary: 53E20

References

Publication

Received: 28 November 2020

Accepted: 16 September 2021

Published: 1 May 2023

Proposed: Tobias H Colding

Seconded: John Lott, Bruce Kleiner

Authors

Simon Brendle
Department of Mathematics Columbia University New York, NY United States

Keaton Naff
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

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Volume 27, issue 1 (2023)