Paper · Aug 11, 2018

Collapsed Manifolds With Ricci Bounded Covering Geometry

Hongzhi Huang, Lingling Kong, Xiaochun Rong, Shicheng Xu

math.DGarXiv:1808.03774
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Abstract

We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.

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Submitted
Aug 11, 2018
Subjects
math.DG
Ricci flow(Non)collapsingHomogeneous spaces

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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 projected homogeneous Ricci flow and three-isotropy-summands flag manifolds (Lino Grama, Ricardo M. Martins, Mauro Patrão et al., 2020)5Grama 2020 On the Ricci Flow of Homogeneous Metrics on Spheres (Sammy Sbiti, 2020)7Sbiti 2020 Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing (Shaosai Huang, Xiaochun Rong and Bing Wang, 2020)2Huang 2020 Collapsed manifolds with local Ricci bounded covering geometry (Xiaochun Rong, 2022)1Rong 2022 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)6Krishnan 2023 Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration (Hanci Chi, 2024)3Chi 2024 Noncollapsing for Curvature Flows with Inhomogeneous Speeds (Weimin Sheng and Ye Zhu, 2025)8Sheng 2025 The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers (Shaosai Huang, 2026)4Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsed manifolds with local Ricci bounded covering geometry

    Xiaochun Rong

    Topics (Non)collapsing, Homogeneous spaces · Same author Xiaochun Rong · Shared terms “covering geometry”, “bounded covering”, “Ricci bounded”

  2. Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

    Shaosai Huang, Xiaochun Rong and Bing Wang

    Topics (Non)collapsing, Homogeneous spaces · Same author Xiaochun Rong · Shared terms “covering geometry”, “collapsed”, “Fukaya”

  3. Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration

    Hanci Chi

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “collapsed”

  4. The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers

    Shaosai Huang

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “Reifenberg”, “Gromov”

  5. The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds

    Lino Grama, Ricardo M. Martins, Mauro Patrão et al.

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “Gromov”

  6. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “collapsed”, “Gromov”

  7. On the Ricci Flow of Homogeneous Metrics on Spheres

    Sammy Sbiti

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “collapsed”

  8. Noncollapsing for Curvature Flows with Inhomogeneous Speeds

    Weimin Sheng and Ye ZhuWider flows

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “structural”

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) (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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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.