Paper · Apr 21, 2020

Rigidity of the first Betti number via Ricci flow smoothing

Shaosai Huang, Bing Wang

math.DGarXiv:2004.09762v3
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Abstract

The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci curved manifold with unit diameter and maximal first Betti number is homeomorphic to the flat torus. In this paper, we prove a parametrized version of this theorem, in the context of collapsing Riemannian manifolds with Ricci curvature bounded below: if a closed manifold with Ricci curvature uniformly bounded below is Gromov-Hausdorff close to a (lower dimensional) manifold with bounded geometry, and has the difference of their first Betti numbers equal to the dimensional difference, then it is diffeomorphic to a torus bundle over the one with bounded geometry. We rely on two novel technical tools: the first is an effective control of the spreading of minimal geodesics with initial data parallel transported along a short geodesic segment, and the second is a Ricci flow smoothing result for certain collapsing initial data with Ricci curvature bounded below.

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Submitted
Apr 21, 2020
Last revised
Dec 30, 2024 (v3)
Subjects
math.DG

Comments: 39 pages, 3 figures. References added

Ricci flow(Non)collapsingLimit spaces & RCDStability & uniqueness

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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 Notes on Ricci flows with collapsing initial data (I): Distance distortion (Shaosai Huang, 2018)4Huang 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)3Huang 2018 Ricci flow smoothing for locally collapsing manifolds (Shaosai Huang and Bing Wang, 2020)1Huang 2020 Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing (Shaosai Huang, Xiaochun Rong and Bing Wang, 2020)2Huang 2020 The local entropy along Ricci flow—Part B: the pseudo-locality theorems (Bing Wang, 2020)7Wang 2020 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)8Li 2021 The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers (Shaosai Huang, 2026)6Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow smoothing for locally collapsing manifolds

    Shaosai Huang and Bing Wang

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

  2. 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”, “collapsing Riemannian”

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

    Shaosai Huang, Yu Li and Bing Wang

    Topics Limit spaces & RCD · Same authors Shaosai Huang, Bing Wang · Shared terms “smoothing”, “Colding”, “Gromov Hausdorff”

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

    Shaosai Huang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Shaosai Huang · Shared terms “collapsing initial”, “initial data”, “uniformly bounded”

  5. On the structure of Ricci shrinkers

    Haozhao Li, Yu Li and Bing Wang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Bing Wang

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

    Shaosai Huang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Shaosai Huang · Shared terms “homeomorphic”, “Gromov”, “Hausdorff”

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

    Bing Wang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Bing Wang · Shared terms “curvature bounded”, “Gromov Hausdorff”

  8. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang

    Topics Limit spaces & RCD, Stability & uniqueness · Same author Bing Wang · Shared terms “Gromov Hausdorff”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.