Paper · Dec 6, 2017

Harnack inequalities on totally geodesic foliations with transverse Ricci flow

Qi Feng

math.DGarXiv:1712.02275v2
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Abstract

In the current paper,under the transverse Ricci flow on a totally geodesic Riemannian foliation, we prove two types of differential Harnack inequalities (Li-Yau gradient estimate) for the positive solutions of the heat equation associated with the time dependent horizontal Laplacian operators. We also get a time dependent version of the generalized curvature dimension inequality. As a consequence of aforementioned results, we also get parabolic Harnack inequalities and heat kernel upper bounds.

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Submitted
Dec 6, 2017
Last revised
Jul 30, 2018 (v2)
Subjects
math.DG

Comments: Notes added and typos corrected

Ricci flowHarnack inequalitiesHeat kernel

Its neighbourhood

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  1. A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

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    Topics Heat kernel, Harnack inequalities · Shared terms “parabolic Harnack”, “positive solutions”, “gradient estimates”

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  4. New differential Harnack inequalities for nonlinear heat equations

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  5. Some new gradient estimates for two nonlinear parabolic equations under Ricci flow

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    Topics Harnack inequalities · Shared terms “positive solutions”, “gradient estimates”, “heat equation”

  6. Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifolds

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  7. On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

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

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    Topics Heat kernel, Harnack inequalities · Shared terms “positive solutions”, “Yau”

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) 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) 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)

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.