Paper · Oct 27, 2026

$O (3)$ -symmetry of four-dimensional steady Ricci solitons

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Abstract

We prove that a complete connected four-dimensional gradient steady Ricci soliton with positive curvature operator that is $κ$ -noncollapsed at all scales admits an effective isometric $O (3)$ action preserving the potential function and having principal orbits diffeomorphic to $S^{2}$ .

Ricci solitons (Non)collapsing 4-manifolds Sphere theorems

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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.

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Topics 4-manifolds, Ricci solitons · Shared terms “steady Ricci”, “dimensional gradient”, “four dimensional”
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Topics 4-manifolds, Ricci solitons · Shared terms “four dimensional”, “gradient steady”, “dimensional gradient”
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Topics Sphere theorems, Ricci solitons · Shared terms “positive curvature”, “curvature operator”
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Topics 4-manifolds, Ricci solitons · Shared terms “dimensional steady”, “four dimensional”
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Topics 4-manifolds, (Non)collapsing, Ricci solitons · Shared terms “dimensional steady”, “steady Ricci”, “four dimensional”
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Topics 4-manifolds, (Non)collapsing · Same author Hanbing Fang · Shared terms “four dimensional”, “noncollapsed”
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Topics 4-manifolds, Ricci solitons · Shared terms “steady Ricci”, “gradient steady”

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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) (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) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.
9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

$O (3)$ -symmetry of four-dimensional steady Ricci solitons

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O(3)-symmetry of four-dimensional steady Ricci solitons

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$O (3)$ -symmetry of four-dimensional steady Ricci solitons