Paper · Oct 21, 2017

Higher dimensional steady Ricci solitons with linear curvature decay

Yuxing Deng, Xiaohua Zhu

math.DGarXiv:1710.07815v2
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Abstract

We prove that any noncompact -noncollapsed steady Ricci soliton with nonnegative curvature operator must be rotationally symmetric if it has a linear curvature decay.

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Submitted
Oct 21, 2017
Last revised
Nov 6, 2017 (v2)
Subjects
math.DG

Comments: We improve the main results in the previous version of paper without the assumption of positive Ricci curvature

Ricci solitons(Non)collapsingBoundaries & non-compactRotational 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)4Deng 2016 3d steady Gradient Ricci Solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2016)1Deng 2016 Classification of gradient steady Ricci solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2018)2Deng 2018 On a dichotomy of the curvature decay of steady Ricci soliton (Pak-Yeung Chan and Bo Zhu, 2021)7Chan 2021 4d steady gradient Ricci solitons with nonnegative curvature away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2023)5Zhao 2023 Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2024)6Zhao 2024 Rigidity of positively curved Steady gradient Ricci solitons on orbifolds (Yuxing Deng, 2025)8Deng 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 3d steady Gradient 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 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, Ricci solitons · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “steady Ricci”, “nonnegative curvature”, “curvature operator”

  4. 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 steady”

  5. steady gradient Ricci solitons with nonnegative curvature away from a compact set

    Ziyi Zhao and Xiaohua Zhu

    Topics (Non)collapsing, Boundaries & non-compact, Ricci solitons · Same author Xiaohua Zhu · Shared terms “nonnegative curvature”, “noncollapsed steady”, “curvature operator”

  6. Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set

    Ziyi Zhao and Xiaohua Zhu

    Topics (Non)collapsing, Boundaries & non-compact, Ricci solitons · Same author Xiaohua Zhu · Shared terms “nonnegative curvature”, “steady Ricci”, “curvature operator”

  7. On a dichotomy of the curvature decay of steady Ricci soliton

    Pak-Yeung Chan and Bo Zhu

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “curvature decay”, “steady Ricci”, “linear curvature”

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

    Yuxing Deng

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

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.