Paper · Jun 13, 2021

Ancient Ricci flows with asymptotic solitons

Pak-Yeung Chan, Zilu Ma, Yongjia Zhang

math.DGarXiv:2106.06904
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Abstract

We study ancient Ricci flows which admit asymptotic solitons in the sense of Perelman. We prove that the asymptotic solitons must coincide with Bamler's tangent flows at infinity. Furthermore, we show that Perelman's -functional is uniformly bounded on such ancient solutions; this fact leads to logarithmic Sobolev inequalities and Sobolev inequalities. Lastly, as an important tool for the proofs of the above results, we also show that, for a complete Ricci flow with bounded curvature, the bound of the Nash entropy depends only on the local geometry around an -center of its base point.

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Submitted
Jun 13, 2021
Subjects
math.DG
Ricci flowAncient solutionsSingularitiesPerelman entropy

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 Perelman's entropy on ancient Ricci flows (Zilu Ma and Yongjia Zhang, 2021)3Ma 2021 A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)1Chan 2021 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)5Chan 2021 An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons (Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al., 2021)6Bamler 2021 A local Sobolev inequality on Ricci flow and its applications (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)2Chan 2021 Local smooth convergence of F-limit flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)4Chan 2023 A rigidity result for ancient Ricci flows (Qi S. Zhang, 2024)7Zhang 2024 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)8Deng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy, Ancient solutions · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “Sobolev inequality”, “Nash entropy”, “logarithmic Sobolev”

  2. A local Sobolev inequality on Ricci flow and its applications

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “Sobolev inequality”, “local geometry”, “Nash entropy”

  3. Perelman's entropy on ancient Ricci flows

    Zilu Ma and Yongjia Zhang

    Topics Perelman entropy, Ancient solutions · Same authors Zilu Ma, Yongjia Zhang

  4. Local smooth convergence of -limit flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “Nash entropy”, “Bamler”, “uniformly bounded”

  5. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Ancient solutions, Singularities · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “tangent flow”, “Bamler”

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

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

    Topics Perelman entropy, Ancient solutions · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “Nash entropy”

  7. A rigidity result for ancient Ricci flows

    Qi S. Zhang

    Topics Perelman entropy, Ancient solutions, Singularities · Shared terms “Sobolev”

  8. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang

    Topics Ancient solutions, Singularities · Same author Yongjia Zhang · Shared terms “tangent flow”

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) Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

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

    Necks and caps, compactness, and universal estimates.

  3. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.