Paper · Aug 6, 2020

Sharp upper diameter bounds for compact shrinking Ricci solitons

Jia-Yong Wu

math.DGarXiv:2008.02893v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

View PDF View HTML arXiv page
Submitted
Aug 6, 2020
Last revised
Feb 22, 2021 (v2)
Subjects
math.DG

Comments: 13 pages, coefficient of Theorem 1.1 improved, accepted by AGAG

Ricci solitonsPerelman entropyScalar curvature

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 Elliptic gradient estimates for a nonlinear heat equation and applications (Jia-Yong Wu, 2016)3Wu 2016 Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures (Michele Rimoldi and Giona Veronelli, 2016)6Rimoldi 2016 Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons (Jia-Yong Wu and Peng Wu, 2019)2Wu 2019 Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)4Wu 2020 Geometric inequalities and rigidity of gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)1Wu 2020 Ancient Ricci flows with asymptotic solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)8Chan 2021 A local Sobolev inequality on Ricci flow and its applications (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)7Chan 2021 Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds (Robin Neumayer, 2022)5Neumayer 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometric inequalities and rigidity of gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “shrinking Ricci”, “compact shrinking”, “Sobolev inequality”

  2. Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

    Jia-Yong Wu and Peng Wu

    Topics Scalar curvature, Ricci solitons · Same author Jia-Yong Wu · Shared terms “shrinking Ricci”, “gradient shrinking”

  3. Elliptic gradient estimates for a nonlinear heat equation and applications

    Jia-Yong Wu

    Topics Perelman entropy, Ricci solitons · Same author Jia-Yong Wu · Shared terms “Sobolev”

  4. Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “sharp upper”, “shrinking Ricci”, “gradient shrinking”

  5. Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds

    Robin Neumayer

    Topics Perelman entropy, Scalar curvature

  6. Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures

    Michele Rimoldi and Giona Veronelli

    Topics Perelman entropy, Ricci solitons · Shared terms “Sobolev inequality”, “shrinking Ricci”

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

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy · Shared terms “Sobolev inequality”

  8. Ancient Ricci flows with asymptotic solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy · Shared terms “Sobolev inequality”, “logarithmic 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 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) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 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.

  2. 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.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.