Paper · Apr 25, 2018

Parabolic frequency monotonicity on compact manifolds

Xiaolong Li, Kui Wang

math.DGarXiv:1804.09806
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Abstract

This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by L. Ni. The proof is based on a generalization of R. Hamilton's matrix Harnack inequality for small time. As applications, we obtain a unique continuation result. Monotonicity of a new quantity under two-dimensional Ricci flow, closely related to the parabolic frequency functional, is derived as well.

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Submitted
Apr 25, 2018
Subjects
math.DG, math.AP

Comments: 17 pages

Ricci flowHarnack inequalitiesSurfaces

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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 2016, 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 New differential Harnack inequalities for nonlinear heat equations (Jia-Yong Wu, 2018)5Wu 2018 Parabolic frequency on Ricci flows (Julius Baldauf and Dain Kim, 2022)6Baldauf 2022 Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures (Chuanhuan Li, Yi Li and Kairui Xu, 2022)7Li 2022 Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency (Xiaolong Li and Qi S. Zhang, 2023)1Li 2023 Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow (Xiaolong Li, Hao-Yue Liu and Xin-An Ren, 2023)8Li 2023 Parabolic frequency monotonicity on the conformal Ricci flow (Abimbola Abolarinwa and Shahroud Azami, 2023)4Abolarinwa 2023 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)2Li 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
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

    Xiaolong Li and Qi S. Zhang

    Topics Harnack inequalities · Same author Xiaolong Li · Shared terms “parabolic frequency”, “unique continuation”, “heat equation”

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

    Chuanhuan Li, Yi Li, Kairui Xu et al.

    Topics Harnack inequalities · Shared terms “parabolic frequency”, “frequency monotonicity”

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

    Juanling Lu and Yu Zheng

    Topics Harnack inequalities · Shared terms “parabolic frequency”, “frequency monotonicity”, “heat equation”

  4. Parabolic frequency monotonicity on the conformal Ricci flow

    Abimbola Abolarinwa and Shahroud Azami

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

  5. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Topics Harnack inequalities, Surfaces · Shared terms “matrix Harnack”, “heat equation”

  6. Parabolic frequency on Ricci flows

    Julius Baldauf and Dain Kim

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

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

    Chuanhuan Li, Yi Li and Kairui Xu

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

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

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

    Topics Harnack inequalities · Same author Xiaolong Li · Shared terms “heat equation”, “Hamilton”

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) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  3. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.