Paper · Feb 16, 2017

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

Yongjia Zhang

math.DGarXiv:1702.05118v2
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Abstract

In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative curvature operator, finite asymptotic entropy implies kappa-noncollapsing on all scales. This provides an evidence for Perelman's more general assertion that on a complete ancient solution with nonnegative curvature operator, bounded entropy is equivalent to kappa-noncollapsing.

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Submitted
Feb 16, 2017
Last revised
Jun 6, 2017 (v2)
Subjects
math.DG
Ricci flowAncient solutions(Non)collapsing

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 On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)1Zhang 2017 The rigidity of S³×ℝ under ancient Ricci flow (Yongjia Zhang, 2017)8Zhang 2017 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)3Li 2018 Ancient solutions to the Kähler Ricci flow (Yu Li, 2020)4Li 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)5Cheng 2021 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)7Deng 2026 A gap theorem for metric solitons and its applications (Ganqi Wang and Yongjia Zhang, 2026)6Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On equivalence between noncollapsing and bounded entropy 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 “curvature operator”, “nonnegative curvature”, “noncollapsing”

  3. 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”

  4. Ancient solutions to the Kähler Ricci flow

    Yu Li

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

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

    Liang Cheng and Yongjia Zhang

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

  6. A gap theorem for metric solitons and its applications

    Ganqi Wang and Yongjia Zhang

    Topics Ancient solutions · Same author Yongjia Zhang · Shared terms “gap theorem”, “counterpart”

  7. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang

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

  8. The rigidity of under ancient Ricci flow

    Yongjia Zhang

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

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

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

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.