Paper · Jul 28, 2016

Aronson-Bénilan estimates for the fast diffusion equation under the Ricci flow

Huai-Dong Cao, Meng Zhu

math.DGarXiv:1607.08555
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Abstract

We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting.

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Submitted
Jul 28, 2016
Subjects
math.DG, math.AP
Ricci flowHarnack inequalitiesHeat kernelBoundaries & non-compact

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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 · Same author Meng Zhu · Shared terms “Yau”, “positive solutions”, “conjugate heat”

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

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

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

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  4. Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci

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    Topics Harnack inequalities · Shared terms “Aronson”, “Bénilan”, “positive solutions”

  5. Harnack inequalities on totally geodesic foliations with transverse Ricci flow

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

  6. Matrix Li-Yau-Hamilton Estimates for Nonlinear Heat Equations

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  7. Linear stability of compact shrinking Ricci solitons

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    Same authors Huai-Dong Cao, Meng Zhu · Shared terms “Cao”, “Hamilton”

  8. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang

    Topics Heat kernel · Shared terms “conjugate heat”, “positive solutions”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.