Paper · Oct 10, 2019

Monotonicity of functionals along conformal Ricci flow

Fengjiang Li, Peng Lu, Jianhong Wang, Yu Zheng

math.DGarXiv:1910.04347
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Abstract

The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entropy functional we give another proof of the monotonicity by establishing a pointwise formula. Moreover, we show that the increase are strict unless the metrics are Einstein.

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Submitted
Oct 10, 2019
Subjects
math.DG, math.AP

Comments: 9 pages

Ricci flowHeat kernel

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Conformal Ricci flow on asymptotically hyperbolic manifolds

    Peng Lu, Jie Qing and Yu Zheng

    Topics Ricci flow · Same authors Peng Lu, Yu Zheng · Shared terms “conformal Ricci”

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

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

    Reto Buzano and Louis Yudowitz

    Topics Heat kernel · Shared terms “closed manifold”

  4. New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

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    Topics Ricci flow · Same authors Peng Lu, Yu Zheng · Shared terms “closed manifold”

  5. Some local Maximum principles along Ricci Flow

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  6. Scalar curvature, entropy, and generalized Ricci flow

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    Topics Heat kernel · Shared terms “conjugate heat”

  7. 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 · Shared terms “closed manifold”, “positive solutions”, “conjugate heat”

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

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

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

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) 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.