Paper · May 16, 2022

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

Chuanhuan Li, Yi Li, Kairui Xu

math.DGarXiv:2205.07702J Geom Anal 33, 282 (2023)
PDF Abstract page

As on arXiv

Abstract

In this paper, we study the monotonicity of parabolic frequency motivated by under the Ricci flow and the Ricci-harmonic flow on manifolds. Here we consider two cases: one is the monotonicity of parabolic frequency for the solution of linear heat equation with bounded Bakry-Émery Ricci curvature, and another case is the monotonicity of parabolic frequency for the solution of heat equation with bounded Ricci curvature.

Comments: 16 pages

Ricci flowLimit spaces & RCD

Its neighbourhood

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 Parabolic frequency on Ricci flows (Julius Baldauf and Dain Kim, 2022)4Baldauf 2022 Parabolic frequency for the mean curvature flow (Julius Baldauf and Tang-Kai Lee, 2022)7Baldauf 2022 Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition (Ling Wu, 2022)8Wu 2022 Parabolic frequency monotonicity on the conformal Ricci flow (Abimbola Abolarinwa and Shahroud Azami, 2023)2Abolarinwa 2023 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)1Li 2023 Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow (Juanling Lu and Yu Zheng, 2025)3Lu 2025 Ricci-harmonic flow of G₂ and Spin(7)-structures (Shubham Dwivedi, 2026)5Dwivedi 2026 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)6Li 2026
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 for two nonlinear equations under Ricci flow

    Chuanhuan Li, Yi Li, Kairui Xu et al. PDF

    Topics Ricci flow · Same authors Chuanhuan Li, Yi Li, Kairui Xu · Shared terms “parabolic frequency”, “frequency monotonicity”, “bounded Ricci”

  2. Parabolic frequency monotonicity on the conformal Ricci flow

    Abimbola Abolarinwa and Shahroud Azami PDF

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

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

    Juanling Lu and Yu Zheng PDF

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

  4. Parabolic frequency on Ricci flows

    Julius Baldauf and Dain Kim PDF

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

  5. Ricci-harmonic flow of and Spin(7)-structures

    Shubham Dwivedi PDF

    Topics Limit spaces & RCD · Shared terms “harmonic flow”, “Ricci harmonic”, “heat equation”

  6. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li PDF

    Topics Ricci flow · Same authors Chuanhuan Li, Yi Li

  7. Parabolic frequency for the mean curvature flow

    Julius Baldauf and Tang-Kai LeeWider flows PDF

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

  8. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu PDF

    Topics Limit spaces & RCD · Shared terms “heat equation”, “Émery Ricci”, “Bakry Émery”

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) 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.