Paper · Nov 5, 2026

Classification of cohomogeneity one steady gradient Ricci solitons in dimension four and five

Yuxing Deng, Guohao Wang

math.DGarXiv:2611.08206
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Abstract

In this paper, we first prove Appleton's conjecuture on the uniqueness of noncollapsed steady solitons constructed in. We also classify non-Ricci-flat -noncollapsed cohomogeneity one steady gradient Ricci solitons in dimension four and five. After passing to a finite cover, in dimension four, they must be the Bryant soliton or Appleton's noncollapsed 4d steady soliton; in dimension five, they must be the Bryant soliton or solitons constructed by Ivey–Dancer–Wang.

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Submitted
Nov 5, 2026
Subjects
math.DG

Comments: 57 pages

Ricci solitonsAncient solutionsRicci-flat & Calabi–Yau(Non)collapsing4-manifoldsHomogeneous spacesStability & uniqueness

Its neighbourhood

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  1. Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

    Yuxing Deng

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same author Yuxing Deng · Shared terms “steady gradient”, “Bryant soliton”, “gradient Ricci”

  2. -invariant steady gradient Ricci solitons on four-manifolds

    Timothy Buttsworth

    Topics 4-manifolds, (Non)collapsing, Homogeneous spaces · Shared terms “steady gradient”, “Appleton”, “cohomogeneity”

  3. On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

    Yuxing Deng and Yuehan Hao

    Topics (Non)collapsing, Ricci solitons · Same author Yuxing Deng · Shared terms “steady gradient”, “gradient Ricci”, “noncollapsed steady”

  4. Asymptotic Geometry of Four-Dimensional Steady Solitons

    Aprameya Girish Hebbar and Natasa Sesum

    Topics 4-manifolds, Ancient solutions, Ricci solitons · Shared terms “steady solitons”, “solitons constructed”, “dimension four”

  5. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang

    Topics Ricci-flat & Calabi–Yau, (Non)collapsing, Ancient solutions · Same author Yuxing Deng · Shared terms “noncollapsed”

  6. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics 4-manifolds, (Non)collapsing, Ricci solitons · Shared terms “steady solitons”, “dimension four”, “noncollapsed steady”

  7. An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons

    Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al.

    Topics (Non)collapsing, Ancient solutions, Ricci solitons · Shared terms “steady gradient”, “noncollapsed steady”, “Appleton”

  8. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Ricci-flat & Calabi–Yau, Ancient solutions, Ricci solitons · Shared terms “Bryant soliton”, “steady solitons”, “steady gradient”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.