Paper · Aug 5, 2021

Rigidity of spherical product Ricci solitons

Ao Sun, Jonathan J. Zhu

math.DGarXiv:2108.02326v3
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Abstract

We show that is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for , where belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.

Comments: 34 pages; corrected a missing term before (4.5) which does not significantly affect the proof strategy

Ricci flowMean curvature flowRicci solitonsEinstein metricsSmale conjecture & diffeomorphismsStability & uniqueness

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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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)5Bamler 2017 On noncompact warped product Ricci solitons (Valter Borges, 2022)1Borges 2022 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)4Sun 2022 Linear stability of compact shrinking Ricci solitons (Huai-Dong Cao and Meng Zhu, 2023)3Cao 2023 The Smale conjecture and min-max theory (Daniel Ketover and Yevgeny Liokumovich, 2023)7Ketover 2023 A rigidity result for axisymmetric toric Ricci solitons (Shiqiao Zhang, 2024)8Zhang 2024 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)6Chen 2026 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)2Braxton 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On noncompact warped product Ricci solitons

    Valter Borges PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons · Shared terms “product ricci”

  2. A stable self-shrinking Möbius bundle in

    Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu PDF

    Topics Stability & uniqueness, Ricci solitons · Same author Jonathan J. Zhu

  3. Linear stability of compact shrinking Ricci solitons

    Huai-Dong Cao and Meng Zhu PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons · Shared terms “shrinking Ricci”

  4. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue PDF

    Topics Ricci solitons · Same author Ao Sun

  5. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce KleinerThe canon PDF

    Topics Smale conjecture & diffeomorphisms, Stability & uniqueness

  6. Closed mean curvature flows with prescribed tangent flows

    Jingwen Chen, Tang-Kai Lee, Ao Sun et al. PDF

    Topics Ricci solitons · Same author Ao Sun

  7. The Smale conjecture and min-max theory

    Daniel Ketover and Yevgeny Liokumovich PDF

    Topics Smale conjecture & diffeomorphisms

  8. A rigidity result for axisymmetric toric Ricci solitons

    Shiqiao Zhang PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    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.