Paper · Aug 28, 2020

Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

Shaosai Huang, Xiaochun Rong, Bing Wang

math.DGarXiv:2008.12419v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.

View PDF View HTML arXiv page
Submitted
Aug 28, 2020
Last revised
Nov 30, 2020 (v3)
Subjects
math.DG
Journal
SIGMA 16 (2020), 123, 25 pages
DOI
10.3842/SIGMA.2020.123

Comments: 25 pages, dedicated to Misha Gromov on the occasion of his 75th birthday

Ricci flowRicci-flat & Calabi–Yau(Non)collapsingKähler geometryHomogeneous spaces

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 Collapsed Manifolds With Ricci Bounded Covering Geometry (Hongzhi Huang, Lingling Kong, Xiaochun Rong et al., 2018)4Huang 2018 On the regular-convexity of Ricci shrinker limit spaces (Shaosai Huang, Yu Li and Bing Wang, 2018)6Huang 2018 Rigidity of the first Betti number via Ricci flow smoothing (Shaosai Huang and Bing Wang, 2020)2Huang 2020 Ricci flow smoothing for locally collapsing manifolds (Shaosai Huang and Bing Wang, 2020)1Huang 2020 The local entropy along Ricci flow—Part B: the pseudo-locality theorems (Bing Wang, 2020)8Wang 2020 On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows (Yang Li and Valentino Tosatti, 2021)5Li 2021 Collapsed manifolds with local Ricci bounded covering geometry (Xiaochun Rong, 2022)3Rong 2022 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. 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”, “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”, “curvature bounded”, “collapsing Riemannian”

  3. Collapsed manifolds with local Ricci bounded covering geometry

    Xiaochun Rong

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

  4. Collapsed Manifolds With Ricci Bounded Covering Geometry

    Hongzhi Huang, Lingling Kong, Xiaochun Rong et al.

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

  5. On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

    Yang Li and Valentino Tosatti

    Topics Ricci-flat & Calabi–Yau, (Non)collapsing, Kähler geometry · Shared terms “Cheeger”, “collapsed”, “Gromov”

  6. 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”, “Cheeger”, “Gromov”

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

    Shaosai Huang

    Topics (Non)collapsing, Homogeneous spaces · Same author Shaosai Huang · Shared terms “Gromov”

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

    Bing Wang

    Topics (Non)collapsing, Kähler geometry · Same author Bing Wang · Shared terms “curvature bounded”, “Gromov”, “Yau”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.