Paper · Apr 28, 2026

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

Chen Wang, Guoqiang Wu

math.DGarXiv:2604.26163
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Abstract

Let be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation . If its scalar curvature is , Cheng-Zhou proved that it is a finite quotient of . In this note we present an alternative proof by analyzing the asymptotic geometry at infinity.

Comments: arXiv admin note: substantial text overlap with arXiv:2604.23939

Ricci solitons4-manifoldsScalar curvatureBoundaries & non-compact

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

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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) 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.