Paper · Sep 11, 2018

On the structure of Ricci shrinkers

Haozhao Li, Yu Li, Bing Wang

math.DGarXiv:1809.04049v2
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Abstract

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.

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Submitted
Sep 11, 2018
Last revised
Feb 19, 2021 (v2)
Subjects
math.DG

Comments: revised version, to appear in J. Funct. Anal

Ricci solitons(Non)collapsingLimit spaces & RCD

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 The Rigidity of Ricci shrinkers of dimension four (Yu Li and Bing Wang, 2017)7Li 2017 On the regular-convexity of Ricci shrinker limit spaces (Shaosai Huang, Yu Li and Bing Wang, 2018)4Huang 2018 Heat kernel on Ricci shrinkers (Yu Li and Bing Wang, 2019)6Li 2019 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)1Li 2021 Rigidity of complex projective spaces in Ricci shrinkers (Yu Li and Wenjia Zhang, 2022)8Li 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)3Li 2023 On Kähler Ricci shrinker surfaces (Yu Li and Bing Wang, 2023)5Li 2023 Non-collapsing of Ricci shrinkers with bounded curvature (Conghan Dong and Yu Li, 2024)2Dong 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang

    Topics Limit spaces & RCD, Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  2. Non-collapsing of Ricci shrinkers with bounded curvature

    Conghan Dong and Yu Li

    Topics (Non)collapsing, Limit spaces & RCD, Ricci solitons · Same author Yu Li · Shared terms “Ricci shrinkers”

  3. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Limit spaces & RCD, Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  4. On the regular-convexity of Ricci shrinker limit spaces

    Shaosai Huang, Yu Li and Bing Wang

    Topics Limit spaces & RCD, Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  5. On Kähler Ricci shrinker surfaces

    Yu Li and Bing Wang

    Topics Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  6. Heat kernel on Ricci shrinkers

    Yu Li and Bing Wang

    Topics (Non)collapsing, Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  7. The Rigidity of Ricci shrinkers of dimension four

    Yu Li and Bing Wang

    Topics Limit spaces & RCD, Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  8. Rigidity of complex projective spaces in Ricci shrinkers

    Yu Li and Wenjia Zhang

    Topics Limit spaces & RCD, Ricci solitons · Same author Yu Li · Shared terms “Ricci shrinkers”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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.