Paper · Jun 16, 2023

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

Xiaolong Li, Qi S. Zhang

math.DGarXiv:2306.10143v2
PDF Abstract page

As on arXiv

Abstract

In this paper we prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply such estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.

Comments: 39 pages. All comments are welcome. Theorem 1.4 and Section 7 added

Ricci flowHarnack inequalitiesHeat kernel

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 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)6Huang 2021 Parabolic frequency on Ricci flows (Julius Baldauf and Dain Kim, 2022)7Baldauf 2022 Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures (Chuanhuan Li, Yi Li and Kairui Xu, 2022)8Li 2022 A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound (Xingyu Song, Ling Wu and Meng Zhu, 2023)5Song 2023 Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow (Xiaolong Li, Hao-Yue Liu and Xin-An Ren, 2023)1Li 2023 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)3Li 2023 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)4Zhang 2024 Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow (Juanling Lu and Yu Zheng, 2025)2Lu 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 Kähler-Ricci Flow

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

    Topics Heat kernel, Harnack inequalities · Same author Xiaolong Li · Shared terms “Hamilton estimates”, “Yau Hamilton”, “backward conjugate”

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

    Juanling Lu and Yu Zheng PDF

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

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

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

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

  4. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang PDF

    Topics Harnack inequalities · Same author Qi S. Zhang · Shared terms “heat equation”, “Hamilton”, “Yau”

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

    Xingyu Song, Ling Wu and Meng Zhu PDF

    Topics Heat kernel, Harnack inequalities · Shared terms “Yau”, “positive solutions”, “conjugate heat”

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

    Hong Huang PDF

    Topics Heat kernel · Shared terms “conjugate heat”, “positive solutions”

  7. Parabolic frequency on Ricci flows

    Julius Baldauf and Dain Kim PDF

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

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

    Chuanhuan Li, Yi Li and Kairui Xu PDF

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

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. 12A.9
    PseudolocalityBook 12A · Entropy and Noncollapsing

    Almost-Euclidean regions stay controlled, and the Harnack inequality behind it.

  3. 12A.10
    The Proof of PseudolocalityBook 12A · Entropy and Noncollapsing

    Perelman's §10 in detail: the point-picking argument, the conjugate heat kernel at the bad point, the localised entropy, and the contradiction with the almost-Euclidean isoperimetric inequality.