Paper · Oct 29, 2026

Steady conformally Kähler Ricci solitons with isoparametric potential

Ernani Ribeiro, Hung Tran

math.DGarXiv:2610.40994
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Abstract

We classify complete, non-compact, simply connected, irreducible four-dimensional steady gradient Ricci solitons that are globally conformally Kähler and have an isoparametric potential. The steady Ricci solitons are -invariant nuts on , including the Bryant and Cao solitons, and bolts on and on . For each there is exactly one non-collapsed bolt. A key step, valid for Ricci solitons of every sign, is that away from the locally conformally flat case, the conformal factor is aligned with the potential function, which yields cohomogeneity-one symmetry.

Comments: 35 pages, 1 figure

Ricci solitonsKähler–Ricci solitons(Non)collapsing4-manifoldsKähler geometryHomogeneous spacesBoundaries & non-compact

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Connected papers

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 SU(2)-invariant steady gradient Ricci solitons on four-manifolds (Timothy Buttsworth, 2021)5Buttsworth 2021 On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons (Xu Cheng, Ernani Ribeiro and Detang Zhou, 2022)7Cheng 2022 Rigidity of four-dimensional Kähler-Ricci solitons (Xiaodong Cao, Ernani Ribeiro and Hung Tran, 2022)1Cao 2022 Kähler Solitons, Contact Structures, and Isoparametric Functions (Hung Tran, 2023)3Tran 2023 Cohomogeneity two Ricci solitons with sub-Euclidean volume (Benjy Firester and Raphael Tsiamis, 2024)8Firester 2024 On Ricci Solitons with Isoparametric Potential Functions (Hung Tran and Kazuo Yamazaki, 2025)2Tran 2025 Kähler-Ricci solitons with almost maximal symmetry (Ha Tuan Dung, Catherine Searle and Hung Tran, 2026)4Dung 2026 O(3)-symmetry of four-dimensional steady Ricci solitons (Hanbing Fang, 2026)6Fang 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 four-dimensional Kähler-Ricci solitons

    Xiaodong Cao, Ernani Ribeiro and Hung Tran PDF

    Topics Kähler–Ricci solitons, 4-manifolds, Kähler geometry · Same authors Ernani Ribeiro, Hung Tran · Shared terms “four dimensional”, “conformally”, “gradient Ricci”

  2. On Ricci Solitons with Isoparametric Potential Functions

    Hung Tran and Kazuo Yamazaki PDF

    Topics Homogeneous spaces, Ricci solitons · Same author Hung Tran · Shared terms “isoparametric potential”, “potential function”, “cohomogeneity”

  3. Kähler Solitons, Contact Structures, and Isoparametric Functions

    Hung Tran PDF

    Topics Kähler–Ricci solitons, 4-manifolds, Homogeneous spaces · Same author Hung Tran · Shared terms “isoparametric”, “simply connected”, “potential function”

  4. Kähler-Ricci solitons with almost maximal symmetry

    Ha Tuan Dung, Catherine Searle and Hung Tran PDF

    Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same author Hung Tran · Shared terms “cohomogeneity”, “potential function”, “gradient Ricci”

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

    Timothy Buttsworth PDF

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

  6. -symmetry of four-dimensional steady Ricci solitons

    Hanbing Fang PDF

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

  7. On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

    Xu Cheng, Ernani Ribeiro and Detang Zhou PDF

    Topics 4-manifolds, Ricci solitons · Same author Ernani Ribeiro · Shared terms “four dimensional”, “potential function”, “gradient Ricci”

  8. Cohomogeneity two Ricci solitons with sub-Euclidean volume

    Benjy Firester and Raphael Tsiamis PDF

    Topics 4-manifolds, (Non)collapsing, Homogeneous spaces · Shared terms “conformal factor”, “cohomogeneity”, “four dimensional”

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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) (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)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

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

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.