Paper · May 2, 2026

Kähler-Ricci solitons with almost maximal symmetry

Ha Tuan Dung, Catherine Searle, Hung Tran

math.DGarXiv:2605.01557
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Abstract

This paper studies a non-trivial gradient Kähler-Ricci soliton, of complex dimension , with an isometry group of dimension at least . We show that the isometry group acts by cohomogeneity one and, consequently, admits a special ansatz involving a Sasakian model. In complex dimension two, we can actually say more: namely, that every such soliton has maximal symmetry; that is, the isometry group is exactly of dimension . In addition, we prove that, if the isometry group acts by cohomogeneity one on a non-trivial gradient Ricci soliton (not necessarily Kähler), the potential function is invariant by the action.

Comments: 21 pages, comments welcome

Ricci solitonsKähler–Ricci solitonsKähler geometryHomogeneous spacesSasakian & contact geometry

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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 Infinitesimal maximal symmetry and Ricci soliton solvmanifolds (Carolyn Gordon and Michael Jablonski, 2021)5Gordon 2021 Rigidity of four-dimensional Kähler-Ricci solitons (Xiaodong Cao, Ernani Ribeiro and Hung Tran, 2022)7Cao 2022 Kähler Gradient Ricci Solitons with Large Symmetry (Hung Tran, 2023)2Tran 2023 Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds (Roberto Mossa and Giovanni Placini, 2023)8Mossa 2023 Kähler Solitons, Contact Structures, and Isoparametric Functions (Hung Tran, 2023)3Tran 2023 On isometry groups of gradient Ricci solitons (Ha Tuan Dung and Hung Tran, 2024)1Dung 2024 On Ricci Solitons with Isoparametric Potential Functions (Hung Tran and Kazuo Yamazaki, 2025)6Tran 2025 Steady conformally Kähler Ricci solitons with isoparametric potential (Ernani Ribeiro and Hung Tran, 2026)4Ribeiro 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On isometry groups of gradient Ricci solitons

    Ha Tuan Dung and Hung Tran PDF

    Topics Ricci solitons · Same authors Ha Tuan Dung, Hung Tran · Shared terms “isometry group”, “trivial gradient”, “gradient Ricci”

  2. Kähler Gradient Ricci Solitons with Large Symmetry

    Hung Tran PDF

    Topics Sasakian & contact geometry, Kähler geometry, Ricci solitons · Same author Hung Tran · Shared terms “isometry”, “gradient Ricci”

  3. Kähler Solitons, Contact Structures, and Isoparametric Functions

    Hung Tran PDF

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

  4. Steady conformally Kähler Ricci solitons with isoparametric potential

    Ernani Ribeiro and Hung Tran PDF

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

  5. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski PDF

    Topics Homogeneous spaces, Ricci solitons · Shared terms “maximal symmetry”, “isometry group”

  6. On Ricci Solitons with Isoparametric Potential Functions

    Hung Tran and Kazuo Yamazaki PDF

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

  7. Rigidity of four-dimensional Kähler-Ricci solitons

    Xiaodong Cao, Ernani Ribeiro and Hung Tran PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Hung Tran · Shared terms “gradient Ricci”

  8. Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds

    Roberto Mossa and Giovanni Placini PDF

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Homogeneous spaces · Shared terms “isometry”

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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 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.