Paper · May 26, 2025
On steady and expanding Ricci solitons with asymptotic symmetries
As on arXiv
Abstract
We establish a symmetry principle for asymptotically cylindrical steady gradient Ricci solitons (GRSs) and asymptotically conical expanding GRSs with homogeneous links. Using this, we show that the Bryant steady soliton is the unique asymptotically cylindrical steady GRS that has a round spherical link and satisfies a particular quantitative rigidity condition. A similar characterization is proved for Bryant's expanding solitons. Finally, we establish a global symmetry result for GRSs which exhibit the aforementioned asymptotics with quotient-Berger sphere asymptotic links.
- Submitted
- May 26, 2025
- Last revised
- Apr 8, 2026 (v3)
- Subjects
- math.DG
Comments: 29 pages. v3 is a substantial rewrite of the paper. Versions 1 and 2 contain a fatal error in Lemma 3.11 (old numbering), so the main claims of those versions are invalid as stated. This version gives a revised symmetry result and includes results for expanders
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Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.