Paper · Nov 10, 2024

Some rigidity results on shrinking gradient Ricci soliton

Jianyu Ou, Yuanyuan Qu, Guoqiang Wu

math.DGarXiv:2411.06395
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Abstract

Suppose is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature is isometric to a finite quotient of , giving a new proof of the main results of Cheng-Zhou.

Ricci solitonsScalar curvatureStability & uniqueness

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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 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2024)2Li 2024 Rigidity of Five-Dimensional shrinking gradient Ricci solitons (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2025)5Li 2025 A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)3Wang 2026 A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)4Wang 2026 Rigidity of shrinking gradient ricci soliton with constant scalar curvature (Fengjiang Li, Yuanyuan Qu and Guoqiang Wu, 2026)1Li 2026 Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid (Wenqi Li, Jianyu Ou, Guoqiang Wu et al., 2026)8Li 2026 Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature (Matheus Horácio, Wenqi Li, Jianyu Ou et al., 2026)7Horácio 2026 Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set (Fengjiang Li, Wenqi Li, Jianyu Ou et al., 2026)6Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same authors Yuanyuan Qu, Guoqiang Wu · Shared terms “shrinking gradient”, “gradient Ricci”, “constant scalar”

  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, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Yuanyuan Qu, Guoqiang Wu · Shared terms “shrinking gradient”, “gradient Ricci”, “constant scalar”

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

    Chen Wang and Guoqiang Wu PDF

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same author Guoqiang Wu · Shared terms “Cheng Zhou”, “shrinking gradient”, “finite quotient”

  4. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu PDF

    Topics Scalar curvature, Ricci solitons · Same author Guoqiang Wu · Shared terms “Cheng Zhou”, “shrinking gradient”, “Zhou Cheng”

  5. Rigidity of Five-Dimensional shrinking gradient Ricci solitons

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

    Topics Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Yuanyuan Qu, Guoqiang Wu · Shared terms “shrinking gradient”, “gradient Ricci”, “complete shrinking”

  6. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al. PDF

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Yuanyuan Qu, Guoqiang Wu · Shared terms “shrinking gradient”, “constant scalar”, “gradient Ricci”

  7. Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

    Matheus Horácio, Wenqi Li, Jianyu Ou et al. PDF

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “constant scalar”, “Petersen Wylie”, “gradient Ricci”

  8. Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al. PDF

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “constant scalar”, “gradient 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)

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.