Paper · May 24, 2017

-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons

Shaosai Huang

math.DGarXiv:1705.08886v3
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Abstract

A closed four dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small -norm of the curvature. In this paper, we localize this fact in the case of shrinking Ricci solitons by proving an -regularity theorem, thus confirming a conjecture of Cheeger-Tian. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four dimensional shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

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Submitted
May 24, 2017
Last revised
Sep 6, 2018 (v3)
Subjects
math.DG
Journal
Int. Math. Res. Not. 2018
DOI
10.1093/imrn/rny069
Ricci solitons(Non)collapsing4-manifoldsBoundaries & non-compact

Its neighbourhood

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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 Structure at infinity for shrinking Ricci solitons (Ovidiu Munteanu and Jiaping Wang, 2016)8Munteanu 2016 A compactness theorem for four-dimensional shrinking gradient Ricci solitons (Yongjia Zhang, 2017)1Zhang 2017 On the classification of four-dimensional gradient Ricci solitons (Fei Yang and Liangdi Zhang, 2017)3Yang 2017 Rigidity of positively curved shrinking Ricci solitons in dimension four (Giovanni Catino, 2018)5Catino 2018 Four-dimensional complete gradient shrinking Ricci solitons (Huai-Dong Cao, Ernani Ribeiro and Detang Zhou, 2020)6Cao 2020 Rigidity of four-dimensional Gradient shrinking Ricci solitons (Xu Cheng and Detang Zhou, 2021)4Cheng 2021 4d steady gradient Ricci solitons with nonnegative curvature away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2023)7Zhao 2023 A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)2Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A compactness theorem for four-dimensional shrinking gradient Ricci solitons

    Yongjia Zhang

    Topics 4-manifolds, (Non)collapsing, Boundaries & non-compact · Shared terms “dimensional shrinking”, “four dimensional”, “closed four”

  2. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “dimensional shrinking”, “four dimensional”, “shrinking Ricci”

  3. On the classification of four-dimensional gradient Ricci solitons

    Fei Yang and Liangdi Zhang

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “four dimensional”, “shrinking Ricci”

  4. Rigidity of four-dimensional Gradient shrinking Ricci solitons

    Xu Cheng and Detang Zhou

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “shrinking Ricci”, “four dimensional”

  5. Rigidity of positively curved shrinking Ricci solitons in dimension four

    Giovanni Catino

    Topics 4-manifolds, Ricci solitons · Shared terms “shrinking Ricci”, “dimensional shrinking”, “four dimensional”

  6. Four-dimensional complete gradient shrinking Ricci solitons

    Huai-Dong Cao, Ernani Ribeiro and Detang Zhou

    Topics 4-manifolds, Ricci solitons · Shared terms “shrinking Ricci”, “four dimensional”

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

    Ziyi Zhao and Xiaohua Zhu

    Topics 4-manifolds, (Non)collapsing, Boundaries & non-compact

  8. Structure at infinity for shrinking Ricci solitons

    Ovidiu Munteanu and Jiaping Wang

    Topics 4-manifolds, Ricci solitons · Shared terms “shrinking Ricci”, “four dimensional”, “bounded 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) (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) 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) 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)

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.