As on arXiv
Abstract
We prove that a steady gradient Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton , where is Ricci flat, or isometric to the Bryant soliton (up to scalings), provided that a couple of geometric conditions inspired by the cigar soliton hold. As an application, we prove that any complete non-compact steady Ricci soliton with positive Ricci curvature controlled by the scalar curvature , curvature tensor satisfying and , as , must be the Bryant soliton. Moreover, we prove that any complete steady soliton with positively pinched Ricci curvature must be Ricci flat.
Comments: arXiv admin note: text overlap with arXiv:1107.4591 by other authors
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Rigidity results for shrinking and expanding Ricci solitons
Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Same authors Benedito Leandro, Jeferson Poveda · Shared terms “pinched Ricci”, “potential function”
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On a dichotomy of the curvature decay of steady Ricci soliton
Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact, Ricci solitons · Shared terms “steady Ricci”, “steady gradient”, “potential function”
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Asymptotic Geometry of Four-Dimensional Steady Solitons
Topics Ancient solutions, Scalar curvature, Boundaries & non-compact · Shared terms “steady solitons”, “Bryant soliton”, “steady gradient”
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Rigidity of positively curved Steady gradient Ricci solitons on orbifolds
Topics Ancient solutions, Scalar curvature, Ricci solitons · Shared terms “Bryant soliton”, “steady gradient”, “complete steady”
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O(2)-symmetry of 3D steady gradient Ricci solitons
Topics Ancient solutions, Ricci solitons · Shared terms “Bryant soliton”, “steady gradient”, “gradient Ricci”
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Is a complete Riemannian manifold with positively pinched Ricci curvature compact
Topics Ancient solutions, Ricci solitons · Shared terms “positively pinched”, “pinched Ricci”, “steady solitons”
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Ancient Ricci flows with nonnegative Ricci curvature
Topics Ricci-flat & Calabi–Yau, Ancient solutions · Shared terms “positively pinched”, “pinched Ricci”
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Infinitely many non-collapsed steady Ricci solitons on complex line bundles
Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact, Ricci solitons · Shared terms “steady Ricci”
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
- 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.
- 11B.2Ancient Solutions and the Harnack Inequality
Comparing curvature across space and time.
- 12B.1κ-Solutions
Ancient, nonnegatively curved, noncollapsed: the models for singularities.