Paper · Oct 17, 2023

Parabolic frequency monotonicity on the conformal Ricci flow

Abimbola Abolarinwa, Shahroud Azami

math.AParXiv:2310.11208v2
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Abstract

This paper is devoted to the investigation of the monotonicity of parabolic frequency functional under conformal Ricci flow defined on a closed Riemannian manifold of constant scalar curvature and dimension not less than 3. Parabolic frequency functional for solutions of certain linear heat equation coupled with conformal pressure is defined and its monotonicity under the conformal Ricci flow is proved by applying Bakry-Emery Ricci curvature bounds. Some consequences of the monotonicity are also presented.

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Submitted
Oct 17, 2023
Last revised
Jan 8, 2024 (v2)
Subjects
math.AP

Comments: 18 pages

Ricci flowScalar curvatureLimit spaces & RCD

Its neighbourhood

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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.

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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. Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures

    Chuanhuan Li, Yi Li and Kairui Xu

    Topics Limit spaces & RCD · Shared terms “parabolic frequency”, “frequency monotonicity”, “linear heat”

  2. Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

    Chuanhuan Li, Yi Li, Kairui Xu et al.

    Topics Ricci flow · Shared terms “parabolic frequency”, “frequency monotonicity”

  3. Parabolic frequency on Ricci flows

    Julius Baldauf and Dain Kim

    Topics Ricci flow · Shared terms “parabolic frequency”, “frequency monotonicity”, “heat equation”

  4. Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow

    Juanling Lu and Yu Zheng

    Topics Ricci flow · Shared terms “parabolic frequency”, “frequency monotonicity”, “heat equation”

  5. Notes on scalar curvature lower bounds of steady gradient Ricci solitons

    Shota Hamanaka

    Topics Limit spaces & RCD, Scalar curvature · Shared terms “Émery Ricci”, “Bakry”, “Riemannian manifolds”

  6. Parabolic frequency for the mean curvature flow

    Julius Baldauf and Tang-Kai LeeWider flows

    Shared terms “parabolic frequency”, “frequency monotonicity”, “heat equation”

  7. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu

    Topics Limit spaces & RCD · Shared terms “Émery Ricci”, “Bakry Émery”, “Riemannian manifolds”

  8. Bounds of Scalar curvature, S-curvature and distortion on -Einstein Finsler manifolds

    Bin Shen

    Topics Limit spaces & RCD, Scalar curvature · Shared terms “curvature bounds”

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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.