Paper · Apr 29, 2024

Kähler Soliton Surfaces Are Generically Toric

Hung Tran

math.DGarXiv:2404.18866v2Calc. Var. 64, 240 (2025)
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Abstract

Let be a Kähler gradient Ricci soliton in real dimension four. One first observes that it is an integrable Hamiltonian system in a classical sense. Indeed, all known complete examples are toric and the symmetry is intrinsically related to the potential function and the scalar curvature . While another article addresses the case that these functions are functionally dependent, this one considers the independent case. The main result states that the soliton admits a toric action under a generic assumption. That is, one assumes that the system is non-degenerate and the potential function is proper. Then there is an effective, completely integrable Hamiltonian toric - action on .

Comments: improve the main statement (using a result of Munteanu-Wang to remove an assumption for the steady case); improve the abstract and Introduction to make clearer the connection to another article

Ricci solitons4-manifoldsSurfacesKähler geometryScalar curvature

Its neighbourhood

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 Rigidity of four-dimensional Kähler-Ricci solitons (Xiaodong Cao, Ernani Ribeiro and Hung Tran, 2022)2Cao 2022 Kähler Gradient Ricci Solitons with Large Symmetry (Hung Tran, 2023)4Tran 2023 Kähler Solitons, Contact Structures, and Isoparametric Functions (Hung Tran, 2023)1Tran 2023 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)3Cao 2024 On Ricci Solitons with Isoparametric Potential Functions (Hung Tran and Kazuo Yamazaki, 2025)8Tran 2025 Kähler-Ricci solitons with almost maximal symmetry (Ha Tuan Dung, Catherine Searle and Hung Tran, 2026)7Dung 2026 Kählerity of complete almost-Kähler gradient shrinking Ricci solitons (Junming Xie, 2026)6Xie 2026 Steady conformally Kähler Ricci solitons with isoparametric potential (Ernani Ribeiro and Hung Tran, 2026)5Ribeiro 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler Solitons, Contact Structures, and Isoparametric Functions

    Hung Tran PDF

    Topics 4-manifolds, Scalar curvature, Kähler geometry · Same author Hung Tran · Shared terms “functionally dependent”, “intrinsically related”, “real dimension”

  2. Rigidity of four-dimensional Kähler-Ricci solitons

    Xiaodong Cao, Ernani Ribeiro and Hung Tran PDF

    Topics 4-manifolds, Scalar curvature, Kähler geometry · Same author Hung Tran · Shared terms “gradient Ricci”

  3. Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

    Xiaodong Cao and Hung Tran PDF

    Topics 4-manifolds, Ricci solitons · Same author Hung Tran · Shared terms “dimension four”, “gradient Ricci”

  4. Kähler Gradient Ricci Solitons with Large Symmetry

    Hung Tran PDF

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

  5. Steady conformally Kähler Ricci solitons with isoparametric potential

    Ernani Ribeiro and Hung Tran PDF

    Topics 4-manifolds, Kähler geometry, Ricci solitons · Same author Hung Tran · Shared terms “potential function”, “gradient Ricci”

  6. Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

    Junming Xie PDF

    Topics 4-manifolds, Kähler geometry, Ricci solitons · Shared terms “Kähler gradient”, “real dimension”, “dimension four”

  7. Kähler-Ricci solitons with almost maximal symmetry

    Ha Tuan Dung, Catherine Searle and Hung Tran PDF

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

  8. On Ricci Solitons with Isoparametric Potential Functions

    Hung Tran and Kazuo Yamazaki PDF

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

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.