Paper · Jun 24, 2024

A rigidity result for ancient Ricci flows

Qi S. Zhang

math.DGarXiv:2406.17179
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Abstract

Using a size condition of the sharp log Sobolev functional (log entropy) near infinity only, we prove a rigidity result for ancient Ricci flows without sign condition on the curvatures. The result is also related to the problem of identifying type II ancient Ricci flows and their backward limits.

Ricci flowAncient solutionsSingularitiesPerelman entropyStability & uniqueness

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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 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)4Chan 2021 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)2Bamler 2021 A local Sobolev inequality on Ricci flow and its applications (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)8Chan 2021 Bochner formulas, functional inequalities and generalized Ricci flow (Eva Kopfer and Jeffrey Streets, 2022)6Kopfer 2022 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)3Bourni 2023 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)5Deng 2026 Uniqueness of positively curved ancient Ricci flows on surfaces with boundary (Kyeongho Bang, Eric Chen and Wenkui Du, 2026)1Bang 2026 Parametrization of ancient Ricci flows with smooth compact tangent shrinkers (Yu Li, 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

    Kyeongho Bang, Eric Chen and Wenkui Du PDF

    Topics Ancient solutions, Singularities, Stability & uniqueness · Shared terms “backward limit”

  2. On the fundamental group of non-collapsed ancient Ricci flows

    Richard H. Bamler PDF

    Topics Ancient solutions, Singularities

  3. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al. PDF

    Topics Ancient solutions · Shared terms “backward”

  4. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities · Shared terms “backward”

  5. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities

  6. Bochner formulas, functional inequalities and generalized Ricci flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Perelman entropy · Shared terms “log Sobolev”

  7. Parametrization of ancient Ricci flows with smooth compact tangent shrinkers

    Yu Li PDF

    Topics Ancient solutions, Singularities · Shared terms “backward”

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

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Perelman entropy · Shared terms “Sobolev”

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) Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… 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.