Paper · May 20, 2017

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

Yongjia Zhang

math.DGarXiv:1705.07240v2
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Abstract

In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the implication in the other direction, that for an ancient solution with bounded nonnegative curvature operator, noncollapsing implies bounded entropy. Hence we prove Perelman's assertion under the assumption of bounded geometry on one time slice. In particular, for ancient solutions of dimension three, we need only to assume bounded curvature. We also establish an equality between the asymptotic entropy and the asymptotic reduce volume, which is a result similar to Xu, where he assumes noncollapsing and the Type I curvature bound.

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Submitted
May 20, 2017
Last revised
Jun 6, 2017 (v2)
Subjects
math.DG, math.AP
Ricci flowAncient solutionsSingularities(Non)collapsing3-manifolds

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 Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)1Zhang 2017 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)4Li 2018 Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow (Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al., 2019)8Angenent 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)6Brendle 2020 Perelman's entropy on ancient Ricci flows (Zilu Ma and Yongjia Zhang, 2021)2Ma 2021 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)3Cheng 2021 Ancient Ricci flows with asymptotic solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)7Chan 2021 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)5Deng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”, “bounded entropy”, “asymptotic entropy”

  2. Perelman's entropy on ancient Ricci flows

    Zilu Ma and Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions · Same author Yongjia Zhang · Shared terms “bounded entropy”, “curvature operator”, “nonnegative curvature”

  3. On the noncollapsedness of positively curved Type I ancient Ricci flows

    Liang Cheng and Yongjia Zhang

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

  4. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

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

  5. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Yongjia Zhang

  6. Uniqueness of compact ancient solutions to three-dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

    Topics (Non)collapsing, Ancient solutions, 3-manifolds · Shared terms “Perelman”

  7. Ancient Ricci flows with asymptotic solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Ancient solutions, Singularities · Same author Yongjia Zhang · Shared terms “bounded curvature”, “Perelman”

  8. Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow

    Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al.

    Topics (Non)collapsing, Ancient solutions, 3-manifolds · Shared terms “Perelman”

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. 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.