Paper · Nov 5, 2026

Scalar curvature bounds and rigidity of Ricci shrinkers

Jianyu Ou, Guoqiang Wu

math.DGarXiv:2611.07381
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Abstract

We prove that a complete noncompact gradient shrinking Ricci soliton of dimension at least two is a finite quotient of an Einstein cylinder if its scalar curvature is bounded below by the cylindrical value. The corresponding pointwise strict bound implies compactness. We also establish one-endedness criteria using curvature information at critical points of the potential, including degenerate ones, and obtain sharp one-endedness and rigidity results under global Ricci norm bounds.

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Submitted
Nov 5, 2026
Subjects
math.DG
Ricci solitonsScalar curvatureLimit spaces & RCDStability & 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 Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application (Weixiong Mai and Jianyu Ou, 2022)8Mai 2022 Some rigidity results on shrinking gradient Ricci soliton (Jianyu Ou, Yuanyuan Qu and Guoqiang Wu, 2024)3Ou 2024 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2024)1Li 2024 Rigidity of Five-Dimensional shrinking gradient Ricci solitons (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2025)6Li 2025 A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)2Wang 2026 A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)7Wang 2026 Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid (Wenqi Li, Jianyu Ou, Guoqiang Wu et al., 2026)4Li 2026 Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set (Fengjiang Li, Wenqi Li, Jianyu Ou et al., 2026)5Li 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 Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics Scalar curvature, Boundaries & non-compact, Stability & uniqueness · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “noncompact gradient”, “finite quotient”, “complete noncompact”

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

    Chen Wang and Guoqiang Wu

    Topics Scalar curvature, Boundaries & non-compact, Stability & uniqueness · Same author Guoqiang Wu · Shared terms “finite quotient”, “noncompact gradient”, “Ricci shrinkers”

  3. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “finite quotient”

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

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al.

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “Ricci shrinkers”, “Einstein”

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

    Fengjiang Li, Wenqi Li, Jianyu Ou et al.

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “Ricci shrinkers”, “Einstein”

  6. Rigidity of Five-Dimensional shrinking gradient Ricci solitons

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “finite quotient”

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

    Chen Wang and Guoqiang Wu

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Same author Guoqiang Wu · Shared terms “noncompact gradient”, “finite quotient”, “complete noncompact”

  8. Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

    Weixiong Mai and Jianyu Ou

    Topics Scalar curvature, Ricci solitons · Same author Jianyu Ou · Shared terms “Ricci shrinkers”, “gradient shrinking”, “shrinking 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.