Paper · Aug 25, 2020

Superconvexity of the Heat Kernel on Hyperbolic Space

Yongzhe Zhang

math.DGWider flowsarXiv:2008.11240
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As on arXiv

Abstract

We prove a conjecture of Bernstein that the superconvexity of the heat kernel on hyperbolic space holds in all dimensions and, hence, there is an analog of Huisken's monotonicity formula for mean curvature flow in hyperbolic space of all dimensions.

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Submitted
Aug 25, 2020
Subjects
math.DG, math.AP
Mean curvature flowHeat kernel

Its neighbourhood

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  1. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel

  2. Well-posedness of mean curvature flow

    Yongheng HanWider flows

    Topics Heat kernel

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    Topics Heat kernel

  4. Heat kernel on Ricci shrinkers

    Yu Li and Bing Wang

    Topics Heat kernel

  5. Nash Entropy and Heat Kernel on Generalized Ricci Flow

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    Topics Heat kernel

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

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    Topics Heat kernel

  7. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

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  8. Perelman's entropy and heat kernel bounds on RCD spaces

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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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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. 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.