Paper · Feb 6, 2021

On the noncollapsedness of positively curved Type I ancient Ricci flows

Liang Cheng, Yongjia Zhang

math.DGarXiv:2102.03508
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Abstract

In this article, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three-dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.

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Submitted
Feb 6, 2021
Subjects
math.DG
Ricci flowAncient solutionsSingularities(Non)collapsing3-manifoldsSphere theoremsBoundaries & non-compact

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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 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 Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)7Zhang 2017 On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)3Zhang 2017 The rigidity of S³×ℝ under ancient Ricci flow (Yongjia Zhang, 2017)8Zhang 2017 On the κ-solutions of the Ricci flow on noncompact 3-manifolds (Liang Cheng and Anqiang Zhu, 2018)1Cheng 2018 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)2Li 2018 Perelman-type no breather theorem for noncompact Ricci flows (Liang Cheng and Yongjia Zhang, 2020)5Cheng 2020 A no expanding breather theorem for noncompact Ricci flows (Liang Cheng and Yongjia Zhang, 2021)4Cheng 2021 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)6Deng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the solutions of the Ricci flow on noncompact 3-manifolds

    Liang Cheng and Anqiang Zhu

    Topics Sphere theorems, 3-manifolds, Boundaries & non-compact · Same author Liang Cheng · Shared terms “dimensional noncompact”, “positive sectional”, “sectional curvature”

  2. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Same author Yongjia Zhang · Shared terms “noncollapsed”

  3. On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow

    Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions, 3-manifolds · Same author Yongjia Zhang · Shared terms “noncollapsing”

  4. A no expanding breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Sphere theorems, Boundaries & non-compact · Same authors Liang Cheng, Yongjia Zhang

  5. Perelman-type no breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, Boundaries & non-compact · Same authors Liang Cheng, Yongjia Zhang

  6. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Yongjia Zhang · Shared terms “noncollapsed”, “positively”

  7. Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

    Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions · Same author Yongjia Zhang · Shared terms “noncollapsing”

  8. The rigidity of under ancient Ricci flow

    Yongjia Zhang

    Topics Ancient solutions, Singularities · Same author Yongjia Zhang · Shared terms “curved Type”

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 can stop in finite time: curvature blows up, 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. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

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

    Necks and caps, compactness, and universal estimates.