Paper · Jul 26, 2022

Scalar curvature, entropy, and generalized Ricci flow

Jeffrey Streets

math.DGarXiv:2207.13197
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Abstract

We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction-diffusion equation motivated by renormalization group flow. These scalar curvature monotonicities are dual to a new family of Perelman-type energy and entropy monotonicity formulas by coupling to a solution of the associated weighted conjugate heat equation. In the setting of Ricci flow, we further obtain a new family of convex Nash entropies and pseudolocality principles.

Ricci flowGeneralized & G₂ flowsHeat kernelScalar curvaturePhysics

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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 2021, 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 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)7Huang 2021 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)6Gindi 2021 Bochner formulas, functional inequalities and generalized Ricci flow (Eva Kopfer and Jeffrey Streets, 2022)4Kopfer 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)2Li 2022 Optimal Transport and Generalized Ricci Flow (Eva Kopfer and Jeffrey Streets, 2023)3Kopfer 2023 Ricci flow on Courant algebroids (Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach, 2024)1Streets 2024 Hermitian curvature flow and HKT geometry (Beatrice Brienza and Jeffrey Streets, 2026)8Brienza 2026 Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)5He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow on Courant algebroids

    Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach PDF

    Topics Generalized & G₂ flows, Physics, Scalar curvature · Same author Jeffrey Streets · Shared terms “curvature monotonicity”, “monotonicity formula”, “Perelman”

  2. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li PDF

    Topics Heat kernel, Generalized & G₂ flows · Shared terms “Nash entropy”

  3. Optimal Transport and Generalized Ricci Flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  4. Bochner formulas, functional inequalities and generalized Ricci flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  5. Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan PDF

    Topics Heat kernel, Generalized & G₂ flows, Scalar curvature · Shared terms “Nash entropy”, “pseudolocality”, “conjugate heat”

  6. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

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

    Hong Huang PDF

    Topics Heat kernel · Shared terms “conjugate heat”

  8. Hermitian curvature flow and HKT geometry

    Beatrice Brienza and Jeffrey Streets PDF

    Topics Generalized & G₂ flows, Scalar curvature · Same author Jeffrey Streets · Shared terms “monotonicity formula”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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.