Paper · Jun 5, 2025

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

Juanling Lu, Yu Zheng

math.DGarXiv:2506.04937
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Abstract

In this paper, we establish Li-Yau-type and Hamilton-type estimates for positive solutions to the heat equation associated with the generalized Ricci flow, under a less stringent curvature condition. Compared with [25] and [35], these estimates generalize the results in Ricci flow to this new flow under the weaker Ricci curvature bounded assumption. As an application, we derive the Harnack-type inequalities in spacetime and find the monotonicity of one parabolic frequency for positive solutions of the heat equation under bounded Ricci curvature.

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Submitted
Jun 5, 2025
Subjects
math.DG
Ricci flowGeneralized & G₂ flowsHarnack inequalities

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)5Baldauf 2022 Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures (Chuanhuan Li, Yi Li and Kairui Xu, 2022)4Li 2022 Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency (Xiaolong Li and Qi S. Zhang, 2023)2Li 2023 A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound (Xingyu Song, Ling Wu and Meng Zhu, 2023)6Song 2023 Parabolic frequency monotonicity on the conformal Ricci flow (Abimbola Abolarinwa and Shahroud Azami, 2023)8Abolarinwa 2023 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)1Li 2023 Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds (Yanlin Li, Abimbola Abolarinwa, Suraj Ghosh et al., 2024)7Li 2024 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)3Zhang 2024
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.

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

  2. Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

    Xiaolong Li and Qi S. Zhang

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

  3. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

    Topics Harnack inequalities · Shared terms “gradient estimate”, “heat equation”, “curvature condition”

  4. 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”

  5. Parabolic frequency on Ricci flows

    Julius Baldauf and Dain Kim

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

  6. A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

    Xingyu Song, Ling Wu and Meng Zhu

    Topics Harnack inequalities · Shared terms “heat equation”, “positive solutions”, “gradient estimate”

  7. Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

    Yanlin Li, Abimbola Abolarinwa, Suraj Ghosh et al.

    Topics Harnack inequalities · Shared terms “positive solutions”, “Harnack type”, “type inequality”

  8. Parabolic frequency monotonicity on the conformal Ricci flow

    Abimbola Abolarinwa and Shahroud Azami

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

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) 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)

In the guidebook

  1. 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.

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

    Comparing curvature across space and time.

  3. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.