Paper · Dec 10, 2022

Rigidity of four-dimensional Kähler-Ricci solitons

Xiaodong Cao, Ernani Ribeiro, Hung Tran

math.DGarXiv:2212.05267
PDF Abstract page

As on arXiv

Abstract

In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.

Ricci solitonsKähler–Ricci solitons4-manifoldsKähler geometryScalar curvatureStability & uniqueness

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 On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons (Xu Cheng, Ernani Ribeiro and Detang Zhou, 2022)5Cheng 2022 Kähler Solitons, Contact Structures, and Isoparametric Functions (Hung Tran, 2023)4Tran 2023 Kähler Soliton Surfaces Are Generically Toric (Hung Tran, 2024)6Tran 2024 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)3Cao 2024 Gradient Shrinking Ricci Solitons and Modified Sectional Curvature (Xiaodong Cao, Ernani Ribeiro and Hosea Wondo, 2025)1Cao 2025 A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)8Wang 2026 A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)7Wang 2026 Steady conformally Kähler Ricci solitons with isoparametric potential (Ernani Ribeiro and Hung Tran, 2026)2Ribeiro 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

    Xiaodong Cao, Ernani Ribeiro and Hosea Wondo PDF

    Topics 4-manifolds, Scalar curvature, Kähler geometry · Same authors Xiaodong Cao, Ernani Ribeiro · Shared terms “four dimensional”, “dual Weyl”, “investigate four”

  2. Steady conformally Kähler Ricci solitons with isoparametric potential

    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”

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

    Xiaodong Cao and Hung Tran PDF

    Topics 4-manifolds, Stability & uniqueness, Ricci solitons · Same authors Xiaodong Cao, Hung Tran · Shared terms “gradient Ricci”

  4. Kähler Solitons, Contact Structures, and Isoparametric Functions

    Hung Tran PDF

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

  5. 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”, “gradient Ricci”

  6. Kähler Soliton Surfaces Are Generically Toric

    Hung Tran PDF

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

  7. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu PDF

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “four dimensional”, “gradient Ricci”, “gradient shrinking”

  8. A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu PDF

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “four dimensional”, “Weyl”, “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) 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. 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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

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

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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