Paper · Dec 17, 2016

3d steady Gradient Ricci Solitons with linear curvature decay

Yuxing Deng, Xiaohua Zhu

math.DGarXiv:1612.05713
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Abstract

In this note, we prove that a 3-dimensional steady Ricci soliton is rotationally symmetric if its scalar curvature satisfies for some constant , where denotes the distance from a fixed point . Our result doesn't assume that the soliton is -noncollapsed.

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Submitted
Dec 17, 2016
Subjects
math.DG
Ricci solitons(Non)collapsing3-manifoldsScalar curvatureRotational symmetry & warped products

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 Steady Ricci solitons with horizontally ε-pinched Ricci curvature (Yuxing Deng and Xiaohua Zhu, 2016)3Deng 2016 Asymptotic behavior of positively curved steady Ricci Solitons, II (Yuxing Deng and Xiaohua Zhu, 2016)5Deng 2016 Level set flow in 3D steady gradient Ricci solitons (Chih-Wei Chen and Kuo-Wei Lee, 2017)7Chen 2017 Higher dimensional steady Ricci solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2017)1Deng 2017 Classification of gradient steady Ricci solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2018)2Deng 2018 A family of 3d steady gradient solitons that are flying wings (Yi Lai, 2020)8Lai 2020 On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature (Yuxing Deng and Yuehan Hao, 2024)6Deng 2024 Rigidity of positively curved Steady gradient Ricci solitons on orbifolds (Yuxing Deng, 2025)4Deng 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Higher dimensional steady Ricci solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

    Topics Rotational symmetry & warped products, (Non)collapsing, Ricci solitons · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “linear curvature”, “curvature decay”, “dimensional steady”

  2. Classification of gradient steady Ricci solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

    Topics 3-manifolds, Ricci solitons · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “linear curvature”, “curvature decay”, “steady Ricci”

  3. Steady Ricci solitons with horizontally -pinched Ricci curvature

    Yuxing Deng and Xiaohua Zhu

    Topics Rotational symmetry & warped products, (Non)collapsing, Scalar curvature · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “steady Ricci”, “rotationally symmetric”, “noncollapsed”

  4. Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

    Yuxing Deng

    Topics (Non)collapsing, Scalar curvature, Ricci solitons · Same author Yuxing Deng · Shared terms “steady gradient”, “linear curvature”, “curvature decay”

  5. Asymptotic behavior of positively curved steady Ricci Solitons, II

    Yuxing Deng and Xiaohua Zhu

    Topics (Non)collapsing, Ricci solitons · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “steady Ricci”, “noncollapsed”

  6. On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

    Yuxing Deng and Yuehan Hao

    Topics (Non)collapsing, Ricci solitons · Same author Yuxing Deng · Shared terms “steady gradient”, “gradient Ricci”, “noncollapsed”

  7. Level set flow in 3D steady gradient Ricci solitons

    Chih-Wei Chen and Kuo-Wei Lee

    Topics 3-manifolds, Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “dimensional steady”

  8. A family of 3d steady gradient solitons that are flying wings

    Yi Lai

    Topics Rotational symmetry & warped products, (Non)collapsing, 3-manifolds · Shared terms “steady gradient”, “dimensional steady”, “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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.