Paper · Sep 27, 2020

Geometric inequalities and rigidity of gradient shrinking Ricci solitons

Jia-Yong Wu

math.DGarXiv:2009.12725v2
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Abstract

In this paper we prove that the Sobolev inequality, the logarithmic Sobolev inequality, the Schrödinger heat kernel upper bound, the Faber-Krahn inequality, the Nash inequality and the Rozenblum-Cwikel-Lieb inequality all equivalently exist on complete gradient shrinking Ricci solitons. We also obtain some integral gap theorems for compact shrinking Ricci solitons.

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Submitted
Sep 27, 2020
Last revised
Jan 16, 2025 (v2)
Subjects
math.DG, math.AP
Journal
Asian Journal of Mathematics, Vol. 28, Issue 4 (2024), pp. 549-580
DOI
10.4310/AJM.250116215812

Comments: Final version

Ricci solitonsHeat kernelStability & uniquenessGeometric 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 Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons (Jia-Yong Wu and Peng Wu, 2019)4Wu 2019 Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)1Wu 2020 Gaussian upper bounds for the heat kernel on evolving manifolds (Reto Buzano and Louis Yudowitz, 2020)3Buzano 2020 Sharp upper diameter bounds for compact shrinking Ricci solitons (Jia-Yong Wu, 2020)2Wu 2020 The sharp Li-Yau equality on Shrinking Ricci Solitons with applications (Jason Ledwidge, 2020)6Ledwidge 2020 On local rigidity theorems with respect to the scalar curvature (Liang Cheng, 2023)5Cheng 2023 Willmore-type inequalities for closed hypersurfaces in weighted manifolds (Guoqiang Wu and Jia-Yong Wu, 2024)7Wu 2024 Sharp Gaussian Isoperimetry along a Ricci Flow (Robert Koirala, 2026)8Koirala 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Heat kernel, Ricci solitons · Same author Jia-Yong Wu · Shared terms “Schrödinger heat”, “shrinking Ricci”, “gradient shrinking”

  2. Sharp upper diameter bounds for compact shrinking Ricci solitons

    Jia-Yong Wu

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

  3. Gaussian upper bounds for the heat kernel on evolving manifolds

    Reto Buzano and Louis Yudowitz

    Topics Heat kernel · Shared terms “upper bound”, “Sobolev inequality”, “logarithmic Sobolev”

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

    Jia-Yong Wu and Peng Wu

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

  5. On local rigidity theorems with respect to the scalar curvature

    Liang Cheng

    Topics Geometric inequalities, Stability & uniqueness · Shared terms “Sobolev inequality”, “logarithmic Sobolev”

  6. The sharp Li-Yau equality on Shrinking Ricci Solitons with applications

    Jason Ledwidge

    Topics Heat kernel, Ricci solitons · Shared terms “shrinking Ricci”, “compact shrinking”

  7. Willmore-type inequalities for closed hypersurfaces in weighted manifolds

    Guoqiang Wu and Jia-Yong Wu

    Topics Geometric inequalities, Ricci solitons · Same author Jia-Yong Wu

  8. Sharp Gaussian Isoperimetry along a Ricci Flow

    Robert Koirala

    Topics Heat kernel, Geometric inequalities, Stability & uniqueness · Shared terms “Sobolev inequality”, “logarithmic”

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) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

In the guidebook

  1. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  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.