Paper · Sep 15, 2020

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

Jason Ledwidge

math.DGarXiv:2009.06951v2
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Abstract

We prove that the sharp Li-Yau equality holds for the conjugate heat kernel on shrinking Ricci solitons without any curvature or volume assumptions. This quantity yields several estimates which allows us to classify four dimensional, non-compact shrinking Ricci solitons, which arise as Type I singularity models to the Ricci flow.

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Submitted
Sep 15, 2020
Last revised
Sep 21, 2020 (v2)
Subjects
math.DG

Comments: 17 pages. The first version claimed the classification was true in all dimensions. More information and results on the conjugate heat kernel given. Removal of asymptotic cone section from the previous version

Ricci flowRicci solitonsSingularitiesHarnack inequalitiesHeat kernel4-manifoldsBoundaries & non-compact

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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 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 Rigidity of positively curved shrinking Ricci solitons in dimension four (Giovanni Catino, 2018)3Catino 2018 Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)5Wu 2020 The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds (Max Hallgren, 2020)4Hallgren 2020 Geometric inequalities and rigidity of gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)2Wu 2020 Rigidity of four-dimensional Gradient shrinking Ricci solitons (Xu Cheng and Detang Zhou, 2021)8Cheng 2021 Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency (Xiaolong Li and Qi S. Zhang, 2023)6Li 2023 A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound (Xingyu Song, Ling Wu and Meng Zhu, 2023)1Song 2023 Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow (Xiaolong Li, Hao-Yue Liu and Xin-An Ren, 2023)7Li 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

    Xingyu Song, Ling Wu and Meng Zhu

    Topics Heat kernel, Harnack inequalities · Shared terms “Yau”, “conjugate heat”

  2. Geometric inequalities and rigidity of gradient shrinking Ricci solitons

    Jia-Yong Wu

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

  3. Rigidity of positively curved shrinking Ricci solitons in dimension four

    Giovanni Catino

    Topics 4-manifolds, Ricci solitons · Shared terms “shrinking Ricci”, “classify four”, “four dimensional”

  4. The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds

    Max Hallgren

    Topics Heat kernel, 4-manifolds, Singularities · Shared terms “conjugate heat”

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

    Jia-Yong Wu

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

  6. Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

    Xiaolong Li and Qi S. Zhang

    Topics Heat kernel, Harnack inequalities · Shared terms “Yau”, “conjugate heat”

  7. Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow

    Xiaolong Li, Hao-Yue Liu and Xin-An Ren

    Topics Heat kernel, Harnack inequalities · Shared terms “Yau”, “conjugate heat”

  8. Rigidity of four-dimensional Gradient shrinking Ricci solitons

    Xu Cheng and Detang Zhou

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “shrinking Ricci”, “four dimensional”

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) 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) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986)

In the guidebook

  1. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  2. 12A.9
    PseudolocalityBook 12A · Entropy and Noncollapsing

    Almost-Euclidean regions stay controlled, and the Harnack inequality behind it.

  3. 12A.10
    The Proof of PseudolocalityBook 12A · Entropy and Noncollapsing

    Perelman's §10 in detail: the point-picking argument, the conjugate heat kernel at the bad point, the localised entropy, and the contradiction with the almost-Euclidean isoperimetric inequality.