Paper · Oct 5, 2021

On the fundamental group of non-collapsed ancient Ricci flows

Richard H. Bamler

math.DGarXiv:2110.02254
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Abstract

We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for -solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a quotient of the fundamental group of the regular part of any tangent flow at infinity.

Comments: 9 pages

Ricci flowAncient solutionsSingularities(Non)collapsing

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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 Structure theory of non-collapsed limits of Ricci flows (Richard H Bamler, 2020)3Bamler 2020 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)5Chan 2021 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)6Pediconi 2021 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)1Krishnan 2023 A rigidity result for ancient Ricci flows (Qi S. Zhang, 2024)8Zhang 2024 On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature (Yuxing Deng and Yuehan Hao, 2024)4Deng 2024 Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture (Richard H. Bamler and Yi Lai, 2025)7Bamler 2025 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)2Deng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti PDF

    Topics (Non)collapsing, Ancient solutions · Shared terms “Collapsed ancient”

  2. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang PDF

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “tangent flow”

  3. Structure theory of non-collapsed limits of Ricci flows

    Richard H BamlerThe canon PDF

    Topics (Non)collapsing, Singularities · Same author Richard H. Bamler · Shared terms “collapsed”, “tangent flow”

  4. On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

    Yuxing Deng and Yuehan Hao PDF

    Topics (Non)collapsing · Shared terms “fundamental group”

  5. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities · Shared terms “tangent flow”, “admitting”

  6. Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

    Francesco Pediconi and Sammy Sbiti PDF

    Topics Ancient solutions · Shared terms “Collapsed ancient”

  7. Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

    Richard H. Bamler and Yi LaiWider flows PDF

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Richard H. Bamler · Shared terms “collapsed”, “tangent flow”

  8. A rigidity result for ancient Ricci flows

    Qi S. Zhang PDF

    Topics Ancient solutions, Singularities

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.