Paper · Apr 1, 2016

Asymptotic behavior of positively curved steady Ricci Solitons, II

Yuxing Deng, Xiaohua Zhu

math.DGarXiv:1604.00142
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Abstract

In this note, we show that any -dimensional -noncollapsed steady Kähler-Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in.

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Submitted
Apr 1, 2016
Subjects
math.DG
Ricci solitonsKähler–Ricci solitons(Non)collapsingKähler geometry

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 3d steady Gradient Ricci Solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2016)4Deng 2016 Higher dimensional steady Ricci solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2017)2Deng 2017 Classification of gradient steady Ricci solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2018)6Deng 2018 4d steady gradient Ricci solitons with nonnegative curvature away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2023)7Zhao 2023 Dimension Reduction for Positively Curved Steady Solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)5Chan 2023 On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature (Yuxing Deng and Yuehan Hao, 2024)8Deng 2024 Rigidity of positively curved Steady gradient Ricci solitons on orbifolds (Yuxing Deng, 2025)1Deng 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 positively curved Steady gradient Ricci solitons on orbifolds

    Yuxing Deng

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

  2. Higher dimensional steady Ricci solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

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

  3. Steady Ricci solitons with horizontally -pinched Ricci curvature

    Yuxing Deng and Xiaohua Zhu

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

  4. 3d steady Gradient Ricci Solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

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

  5. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics (Non)collapsing, Ricci solitons · Shared terms “curved steady”, “positively curved”, “noncollapsed steady”

  6. 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 “steady Ricci”

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

    Ziyi Zhao and Xiaohua Zhu

    Topics (Non)collapsing, Ricci solitons · Same author Xiaohua Zhu · Shared terms “noncollapsed steady”, “asymptotic behavior”

  8. 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 “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) 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) (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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.