Paper · Feb 9, 2021

Four-Dimensional Steady Gradient Ricci Solitons with -Cylindrical Tangent Flows at Infinity

Richard Bamler, Bennett Chow, Yuxing Deng, Zilu Ma, Yongjia Zhang

math.DGarXiv:2102.04649v2
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Abstract

In this paper we consider -dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed -manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of with a -dimensional spherical space form. We also classify the tangent flows at infinity of -dimensional steady soliton singularity models in general.

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Submitted
Feb 9, 2021
Last revised
Mar 18, 2022 (v2)
Subjects
math.DG

Comments: 13 pages

Ricci flowRicci solitonsSingularities4-manifoldsSphere theorems

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 2016, 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 Curvature growth of some 4-dimensional gradient Ricci soliton singularity models (Bennett Chow, Michael Freedman, Henry Shin et al., 2019)2Chow 2019 On Four-dimensional Steady gradient Ricci solitons that dimension reduce (Bennett Chow, Yuxing Deng and Zilu Ma, 2020)1Chow 2020 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)5Chan 2021 Hamilton-Ivey estimates for gradient Ricci solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)6Chan 2021 Lower bounds for the scalar curvatures of Ricci flow singularity models (Pak-Yeung Chan, Bennett Chow, Zilu Ma et al., 2022)7Chan 2022 Dimension Reduction for Positively Curved Steady Solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)3Chan 2023 Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models (Bennett Chow, Michael H. Freedman, Henry Shin et al., 2025)4Chow 2025 O(3)-symmetry of four-dimensional steady Ricci solitons (Hanbing Fang, 2026)8Fang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Four-dimensional Steady gradient Ricci solitons that dimension reduce

    Bennett Chow, Yuxing Deng and Zilu Ma

    Topics Sphere theorems, 4-manifolds, Singularities · Same authors Bennett Chow, Yuxing Deng, Zilu Ma · Shared terms “dimensional steady”, “steady gradient”, “four dimensional”

  2. Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

    Bennett Chow, Michael Freedman, Henry Shin et al.

    Topics 4-manifolds, Singularities, Ricci solitons · Same authors Bennett Chow, Yongjia Zhang · Shared terms “singularity models”, “gradient Ricci”

  3. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics 4-manifolds, Singularities, Ricci solitons · Same authors Zilu Ma, Yongjia Zhang · Shared terms “steady soliton”, “steady gradient”, “tangent flow”

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

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

    Topics 4-manifolds, Singularities, Ricci solitons · Same authors Bennett Chow, Yongjia Zhang · Shared terms “singularity models”, “gradient Ricci”

  5. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Singularities · Same authors Zilu Ma, Yongjia Zhang · Shared terms “tangent flow”, “singularity models”, “finite time”

  6. Hamilton-Ivey estimates for gradient Ricci solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Singularities, Ricci solitons · Same authors Zilu Ma, Yongjia Zhang · Shared terms “dimensional steady”, “steady gradient”, “gradient Ricci”

  7. Lower bounds for the scalar curvatures of Ricci flow singularity models

    Pak-Yeung Chan, Bennett Chow, Zilu Ma et al.

    Topics Singularities, Ricci solitons · Same authors Bennett Chow, Zilu Ma, Yongjia Zhang · Shared terms “singularity models”, “steady soliton”

  8. -symmetry of four-dimensional steady Ricci solitons

    Hanbing Fang

    Topics Sphere theorems, 4-manifolds, Ricci solitons · Shared terms “dimensional steady”, “four dimensional”

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 can stop in finite time: curvature blows up, 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 the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 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.

  3. 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.