Paper · Sep 22, 2018

Classification of gradient steady Ricci solitons with linear curvature decay

Yuxing Deng, Xiaohua Zhu

math.DGarXiv:1809.08502
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Abstract

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional -noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

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Submitted
Sep 22, 2018
Subjects
math.DG
Ricci solitons3-manifolds4-manifolds

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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 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)5Deng 2016 Asymptotic behavior of positively curved steady Ricci Solitons, II (Yuxing Deng and Xiaohua Zhu, 2016)6Deng 2016 3d steady Gradient Ricci Solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2016)1Deng 2016 Higher dimensional steady Ricci solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2017)2Deng 2017 Curvature estimates for steady Ricci solitons (Pak-Yeung Chan, 2019)7Chan 2019 On a dichotomy of the curvature decay of steady Ricci soliton (Pak-Yeung Chan and Bo Zhu, 2021)3Chan 2021 4d steady gradient Ricci solitons with nonnegative curvature away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2023)8Zhao 2023 Unique Asymptotics of Steady Ricci Solitons with Symmetry (Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum, 2023)4Ma 2023
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 3-manifolds, Ricci solitons · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “linear curvature”, “curvature decay”, “steady Ricci”

  2. Higher dimensional 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. On a dichotomy of the curvature decay of steady Ricci soliton

    Pak-Yeung Chan and Bo Zhu

    Topics 4-manifolds, Ricci solitons · Shared terms “curvature decay”, “steady Ricci”, “linear curvature”

  4. Unique Asymptotics of Steady Ricci Solitons with Symmetry

    Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum

    Topics 4-manifolds, 3-manifolds, Ricci solitons · Shared terms “steady Ricci”, “gradient steady”, “curvature decay”

  5. Steady Ricci solitons with horizontally -pinched Ricci curvature

    Yuxing Deng and Xiaohua Zhu

    Topics Ricci solitons · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “steady Ricci”, “gradient steady”

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

    Yuxing Deng and Xiaohua Zhu

    Topics Ricci solitons · Same authors Yuxing Deng, Xiaohua Zhu · Shared terms “steady Ricci”

  7. Curvature estimates for steady Ricci solitons

    Pak-Yeung Chan

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

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

    Ziyi Zhao and Xiaohua Zhu

    Topics 4-manifolds, Ricci solitons · Same author Xiaohua Zhu · Shared terms “nonnegative sectional”, “sectional curvature”

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) 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) 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

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    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.