Paper · Jan 4, 2021

Perelman's entropy on ancient Ricci flows

Zilu Ma, Yongjia Zhang

math.DGarXiv:2101.01233v2
View PDF View HTML arXiv page

As on arXiv

Abstract

In [ZY2], the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature.

View PDF View HTML arXiv page
Submitted
Jan 4, 2021
Last revised
Jul 7, 2021 (v2)
Subjects
math.DG
Ricci flowRicci solitonsAncient solutionsPerelman entropy(Non)collapsingHarnack inequalities

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)4Zhang 2017 On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)3Zhang 2017 Ancient Ricci flows with asymptotic solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)1Chan 2021 A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)2Chan 2021 An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons (Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al., 2021)5Bamler 2021 A local Sobolev inequality on Ricci flow and its applications (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)6Chan 2021 Local smooth convergence of F-limit flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)7Chan 2023 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. Ancient Ricci flows with asymptotic solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

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

  2. 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 Zilu Ma, Yongjia Zhang

  3. 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 “bounded entropy”, “curvature operator”, “nonnegative curvature”

  4. 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 “curvature operator”, “nonnegative curvature”, “noncollapsing”

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

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

    Topics Perelman entropy, (Non)collapsing, Ancient solutions · Same authors Zilu Ma, Yongjia Zhang

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

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy · Same authors Zilu Ma, Yongjia Zhang

  7. Local smooth convergence of -limit flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy · Same authors Zilu Ma, Yongjia Zhang

  8. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang

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

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) 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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 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.

  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.