Paper · Oct 17, 2023

Kähler Solitons, Contact Structures, and Isoparametric Functions

Hung Tran

math.DGarXiv:2310.11328v3
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Abstract

All known examples of simply-connected gradient Kähler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function and the scalar curvature . In this article, we consider the case that and are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.

Comments: Accepted to The Journal of Geometric Analysis with several minor changes from the previous version

Ricci solitonsKähler–Ricci solitons4-manifoldsKähler geometryScalar curvatureHomogeneous spacesSasakian & contact geometry

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)4Cao 2022 Kähler Gradient Ricci Solitons with Large Symmetry (Hung Tran, 2023)5Tran 2023 Kähler Soliton Surfaces Are Generically Toric (Hung Tran, 2024)1Tran 2024 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)7Cao 2024 On Ricci Solitons with Isoparametric Potential Functions (Hung Tran and Kazuo Yamazaki, 2025)6Tran 2025 Generalized η-Ricci solitons on LP-Sasakian manifolds admitting the general connection (Murat Altunbaş and Ayşe Karanlık Akpınar, 2025)8Altunbaş 2025 Kähler-Ricci solitons with almost maximal symmetry (Ha Tuan Dung, Catherine Searle and Hung Tran, 2026)2Dung 2026 Steady conformally Kähler Ricci solitons with isoparametric potential (Ernani Ribeiro and Hung Tran, 2026)3Ribeiro 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler Soliton Surfaces Are Generically Toric

    Hung Tran PDF

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

  2. Kähler-Ricci solitons with almost maximal symmetry

    Ha Tuan Dung, Catherine Searle and Hung Tran PDF

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

  3. Steady conformally Kähler Ricci solitons with isoparametric potential

    Ernani Ribeiro and Hung Tran PDF

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

  4. Rigidity of four-dimensional Kähler-Ricci solitons

    Xiaodong Cao, Ernani Ribeiro and Hung Tran PDF

    Topics Kähler–Ricci solitons, 4-manifolds, Scalar curvature · Same author Hung Tran

  5. Kähler Gradient Ricci Solitons with Large Symmetry

    Hung Tran PDF

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

  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 “isoparametric”, “potential function”, “cohomogeneity”

  7. 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”

  8. Generalized Ricci solitons on LP-Sasakian manifolds admitting the general connection

    Murat Altunbaş and Ayşe Karanlık Akpınar PDF

    Topics Sasakian & contact geometry, 4-manifolds, Ricci solitons

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

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

  2. 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.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.