Paper · Oct 4, 2022

Rigidity results for shrinking and expanding Ricci solitons

Benedito Leandro, Jeferson Poveda

math.DGarXiv:2210.01945
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Abstract

In this paper, we prove some rigidity results for both shrinking and expanding Ricci solitons. First, we prove that compact shrinking Ricci solitons are Einstein if we control the maximum value of the potential function. Then, we prove some rigidity results for non-compact gradient expanding and shrinking Ricci solitons with pinched Ricci and scalar curvatures, assuming an asymptotic condition on the scalar curvature at infinity.

Ricci solitonsScalar curvatureStability & uniquenessBoundaries & non-compact

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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 Curvature estimates for four-dimensional complete gradient expanding Ricci solitons (Huai-Dong Cao and Tianbo Liu, 2021)6Cao 2021 Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio (Huai-Dong Cao, Tianbo Liu and Junming Xie, 2022)4Cao 2022 A comparison theorem for steady Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)1Leandro 2022 New expanding Ricci solitons starting in dimension four (Jan Nienhaus and Matthias Wink, 2023)3Nienhaus 2023 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2024)2Li 2024 On steady and expanding Ricci solitons with asymptotic symmetries (Michael B. Law, 2025)8Law 2025 A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)5Wang 2026 Rigidity of shrinking gradient ricci soliton with constant scalar curvature (Fengjiang Li, Yuanyuan Qu and Guoqiang Wu, 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda PDF

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Same authors Benedito Leandro, Jeferson Poveda · Shared terms “pinched Ricci”, “potential function”

  2. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al. PDF

    Topics Scalar curvature, Boundaries & non-compact, Stability & uniqueness · Shared terms “shrinking Ricci”, “potential function”

  3. New expanding Ricci solitons starting in dimension four

    Jan Nienhaus and Matthias Wink PDF

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Shared terms “expanding Ricci”, “gradient expanding”, “Einstein”

  4. Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

    Huai-Dong Cao, Tianbo Liu and Junming Xie PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “expanding Ricci”, “gradient expanding”

  5. A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu PDF

    Topics Scalar curvature, Boundaries & non-compact, Stability & uniqueness · Shared terms “shrinking Ricci”, “potential function”

  6. Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

    Huai-Dong Cao and Tianbo Liu PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “expanding Ricci”, “gradient expanding”

  7. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons

  8. On steady and expanding Ricci solitons with asymptotic symmetries

    Michael B. Law PDF

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “expanding 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.

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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