Inventiones mathematicae

, Volume 194, Issue 3, pp 731–764 | Cite as

Rotational symmetry of self-similar solutions to the Ricci flow

  • Simon Brendle


Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman’s first paper.


Soliton Vector Field Scalar Curvature Ricci Soliton Ricci Flow 
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It is a pleasure to thank Professors Huai-Dong Cao, Gerhard Huisken, Sergiu Klainerman, Leon Simon, Brian White, for discussions. The author is grateful to Meng Zhu for comments on an earlier version of this paper.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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