Paper · Jan 7, 2024

On noncollapsed -limit metric solitons

Pak-Yeung Chan, Zilu Ma, Yongjia Zhang

math.DGarXiv:2401.03387
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Abstract

A noncollapsed -limit metric soliton is a self-similar singularity model that inevitably arises when studying the Ricci flow with the tool of -convergence [Bam20a,Bam20b,Bam20c]. In this article, we shall present a systematic study of the noncollapsed -limit metric soliton, and show that, apart from the known results in [Bam20c], it satisfies many properties of smooth Ricci shrinkers. In particular, we show a quadratic lower bound for the scalar curvature, a local gap theorem, a global Sobolev inequality, and an optimal volume growth lower bound.

Ricci flowRicci solitonsSingularities(Non)collapsingScalar curvature

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 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 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)8Chan 2021 An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons (Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al., 2021)2Bamler 2021 Lower bounds for the scalar curvatures of Ricci flow singularity models (Pak-Yeung Chan, Bennett Chow, Zilu Ma et al., 2022)3Chan 2022 A local gap theorem for Ricci shrinkers (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2022)7Chan 2022 Dimension Reduction for Positively Curved Steady Solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)6Chan 2023 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)5Deng 2026 Rigidity and gap theorems for Ricci shrinkers (Pak-Yeung Chan and Yongjia Zhang, 2026)4Chan 2026 A gap theorem for metric solitons and its applications (Ganqi Wang and Yongjia Zhang, 2026)1Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A gap theorem for metric solitons and its applications

    Ganqi Wang and Yongjia Zhang PDF

    Topics Scalar curvature, Singularities · Same author Yongjia Zhang · Shared terms “limit metric”, “gap theorem”

  2. An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons

    Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al. PDF

    Topics (Non)collapsing, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “optimal volume”, “volume growth”, “noncollapsed”

  3. Lower bounds for the scalar curvatures of Ricci flow singularity models

    Pak-Yeung Chan, Bennett Chow, Zilu Ma et al. PDF

    Topics Scalar curvature, Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “singularity models”, “lower bound”, “quadratic”

  4. Rigidity and gap theorems for Ricci shrinkers

    Pak-Yeung Chan and Yongjia Zhang PDF

    Topics Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Yongjia Zhang · Shared terms “gap theorem”, “local gap”, “Ricci shrinkers”

  5. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang PDF

    Topics (Non)collapsing, Singularities · Same author Yongjia Zhang · Shared terms “limit metric”, “noncollapsed”

  6. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics (Non)collapsing, Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “noncollapsed”

  7. A local gap theorem for Ricci shrinkers

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “local gap”, “gap theorem”, “Ricci shrinkers”

  8. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Singularities · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “singularity models”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (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.