Paper · Jul 26, 2021

On the improved no-local-collapsing theorem of Ricci flow

Wangjian Jian

math.DGarXiv:2107.12515
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Abstract

In this note we derive an improved no-local-collapsing theorem of Ricci flow under the scalar curvature bound condition along the worldline of the basepoint. It is a refinement of Perelman's no-local-collapsing theorem.

Comments: Comments welcome!

Ricci flow(Non)collapsingScalar curvature

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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 2021, 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 Tangent Flows of Kähler Metric Flows (Max Hallgren and Wangjian Jian, 2022)8Hallgren 2022 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)1Jian 2023 On noncollapsed F-limit metric solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2024)6Chan 2024 Volume estimates and convergence results for solutions to Ricci flow with L^(p) bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)4Liu 2024 Entropy and Non-Collapse in Lorentzian Geometry (Rohit Dhormare, 2026)3Dhormare 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)5Jian 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)7Huang 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)2Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics (Non)collapsing, Scalar curvature · Same author Wangjian Jian · Shared terms “Perelman”

  2. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

    Wangjian Jian and Jian Song PDF

    Topics (Non)collapsing, Scalar curvature · Same author Wangjian Jian

  3. Entropy and Non-Collapse in Lorentzian Geometry

    Rohit Dhormare PDF

    Topics (Non)collapsing · Shared terms “collapsing theorem”, “Perelman”

  4. Volume estimates and convergence results for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon PDF

    Topics (Non)collapsing, Scalar curvature · Shared terms “improved”

  5. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles

    Wangjian Jian and Jian Song PDF

    Topics (Non)collapsing · Same author Wangjian Jian

  6. On noncollapsed -limit metric solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics (Non)collapsing, Scalar curvature

  7. Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics (Non)collapsing, Scalar curvature

  8. Tangent Flows of Kähler Metric Flows

    Max Hallgren and Wangjian Jian PDF

    Topics (Non)collapsing · Same author Wangjian Jian

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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.