Paper · Apr 27, 2026

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

Chen Wang, Guoqiang Wu

math.DGarXiv:2604.23939
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Abstract

Let be an -dimensional complete noncompact gradient shrinking Ricci soliton with the equation . 1. If its scalar curvature is , Ricci curvature is nonnegative and sectional curvature has upper bound , we prove that the Ricci shrinker is isometric to a finite quotient of . 2. If has constant scalar curvature , and each level set of has vanishing Weyl curvature, we prove that it is a finite quotient of . This can be seen a generalization of Cheng-Zhou's four dimensional result to high dimension, since the level set of the potential function has vanishing Weyl curvature automatically when .

Ricci solitons4-manifoldsScalar curvatureStability & uniquenessBoundaries & non-compact

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

    Chen Wang and Guoqiang Wu PDF

    Topics 4-manifolds, Scalar curvature, Boundaries & non-compact · Same authors Chen Wang, Guoqiang Wu · Shared terms “Cheng Zhou”, “constant scalar”, “shrinking gradient”

  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 · Same author Guoqiang Wu · Shared terms “constant scalar”, “shrinking gradient”, “finite quotient”

  3. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu PDF

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

  4. 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 author Guoqiang Wu · Shared terms “constant scalar”, “shrinking gradient”, “gradient Ricci”

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

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  6. Rigidity of Five-Dimensional shrinking gradient Ricci solitons

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

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    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same author Guoqiang Wu · Shared terms “constant scalar”, “Ricci shrinkers”, “gradient Ricci”

  8. A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

    Huai-Dong Cao and Junming Xie PDF

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “Weyl curvature”, “four dimensional”, “dimensional complete”

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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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.