Paper · Aug 23, 2020

Ricci flow smoothing for locally collapsing manifolds

Shaosai Huang, Bing Wang

math.DGarXiv:2008.09956v2
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Abstract

We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.

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Submitted
Aug 23, 2020
Last revised
Sep 1, 2020 (v2)
Subjects
math.DG

Comments: 31 pages, references updated

Ricci flow(Non)collapsing

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 ε-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons (Shaosai Huang, 2017)6Huang 2017 Notes on Ricci flows with collapsing initial data (I): Distance distortion (Shaosai Huang, 2018)3Huang 2018 On the structure of Ricci shrinkers (Haozhao Li, Yu Li and Bing Wang, 2018)5Li 2018 On the regular-convexity of Ricci shrinker limit spaces (Shaosai Huang, Yu Li and Bing Wang, 2018)4Huang 2018 Rigidity of the first Betti number via Ricci flow smoothing (Shaosai Huang and Bing Wang, 2020)2Huang 2020 Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing (Shaosai Huang, Xiaochun Rong and Bing Wang, 2020)1Huang 2020 The local entropy along Ricci flow—Part B: the pseudo-locality theorems (Bing Wang, 2020)8Wang 2020 The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers (Shaosai Huang, 2026)7Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

    Shaosai Huang, Xiaochun Rong and Bing Wang

    Topics (Non)collapsing · Same authors Shaosai Huang, Bing Wang · Shared terms “flow smoothing”, “curvature bounded”

  2. Rigidity of the first Betti number via Ricci flow smoothing

    Shaosai Huang and Bing Wang

    Topics (Non)collapsing · Same authors Shaosai Huang, Bing Wang · Shared terms “flow smoothing”, “collapsing initial”, “curvature bounded”

  3. Notes on Ricci flows with collapsing initial data (I): Distance distortion

    Shaosai Huang

    Topics (Non)collapsing · Same author Shaosai Huang · Shared terms “collapsing initial”, “distance distortion”, “distortion estimate”

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

    Shaosai Huang, Yu Li and Bing Wang

    Same authors Shaosai Huang, Bing Wang · Shared terms “smoothing”

  5. On the structure of Ricci shrinkers

    Haozhao Li, Yu Li and Bing Wang

    Topics (Non)collapsing · Same author Bing Wang

  6. -Regularity and Structure of 4-dimensional Shrinking Ricci Solitons

    Shaosai Huang

    Topics (Non)collapsing · Same author Shaosai Huang · Shared terms “appendix”

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

    Shaosai Huang

    Topics (Non)collapsing · Same author Shaosai Huang · Shared terms “topological”

  8. The local entropy along Ricci flow—Part B: the pseudo-locality theorems

    Bing Wang

    Topics (Non)collapsing · Same author Bing Wang · Shared terms “curvature bounded”

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)

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.