Paper · Sep 19, 2024

Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

Xiaodong Cao, Hung Tran

math.DGarXiv:2409.13123
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Abstract

[Dedicated to Richard S. Hamilton on forty years of Ricci flow] Gradient Ricci solitons have garnered significant attention both as self-similar solutions and singularity models of the Ricci flow. This survey article starts with a list of examples; it also provides some geometric aspects of gradient Ricci solitons, including various asymptotic behaviors; finally, it discusses some recent results on classification and rigidity. In particular, this survey focuses on dimension four.

Comments: survey paper

Ricci flowRicci solitonsSingularities4-manifoldsStability & uniqueness

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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)1Cao 2022 Kähler Solitons, Contact Structures, and Isoparametric Functions (Hung Tran, 2023)7Tran 2023 Kähler Soliton Surfaces Are Generically Toric (Hung Tran, 2024)2Tran 2024 Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four (Cameron MacMahon, 2024)4MacMahon 2024 Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models (Bennett Chow, Michael H. Freedman, Henry Shin et al., 2025)5Chow 2025 Gradient Shrinking Ricci Solitons and Modified Sectional Curvature (Xiaodong Cao, Ernani Ribeiro and Hosea Wondo, 2025)6Cao 2025 On curvature estimates for four-dimensional gradient Ricci solitons (Huai-Dong Cao, 2025)3Cao 2025 Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas (James Siene, 2025)8Siene 2025
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 4-manifolds, Stability & uniqueness, Ricci solitons · Same authors Xiaodong Cao, Hung Tran · Shared terms “gradient Ricci”

  2. Kähler Soliton Surfaces Are Generically Toric

    Hung Tran PDF

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

  3. On curvature estimates for four-dimensional gradient Ricci solitons

    Huai-Dong Cao PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “gradient Ricci”

  4. Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four

    Cameron MacMahon PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “dimension four”

  5. Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

    Bennett Chow, Michael H. Freedman, Henry Shin et al. PDF

    Topics 4-manifolds, Singularities, Ricci solitons · Shared terms “gradient Ricci”, “singularity models”

  6. Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

    Xiaodong Cao, Ernani Ribeiro and Hosea Wondo PDF

    Topics 4-manifolds, Ricci solitons · Same author Xiaodong Cao

  7. Kähler Solitons, Contact Structures, and Isoparametric Functions

    Hung Tran PDF

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

  8. Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas

    James Siene PDF

    Topics 4-manifolds, Stability & uniqueness, 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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.