Paper · Feb 9, 2025

Geometric flows and space-periodic solitons on the light-cone

Yun Yang

math.DGWider flowsarXiv:2502.06035
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Abstract

This paper investigates curve flows on the light-cone in the 3-dimensional Minkowski space. We derive the Harnack inequality for the heat flow and present a detailed classification of space-periodic solitons for a third-order curvature flow. The nontrivial periodic solutions to this flow are expressed in terms of the Jacobi elliptic sine function. Additionally, the closed soliton solutions form a family of transcendental curves, denoted by , which are characterized by a rotation index and close after periods of their curvature functions. The ratio satisfies , where and are relatively prime positive integers. Guided by the classification process, we obtain the analytic solutions to a second-order nonlinear ordinary differential equation.

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Submitted
Feb 9, 2025
Subjects
math.DG
Other curvature flowsHarnack inequalities3-manifolds

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 Self-Similar Solutions to the Curvature Flow and its Inverse on the 2-dimensional Light Cone (Fabio Nunes da Silva and Keti Tenenblat, 2021)1da Silva 2021 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)6Li 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)8Zhang 2024 Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions (Junyoung Park, 2025)3Park 2025 Harnack Inequality for f-Mean Curvature Flow (Xiang-Dong Li and Qi Yan, 2025)2Li 2025 The Harnack inequality without convexity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2026)4Sobnack 2026 Lectures on curve shortening flow (Robert Haslhofer, 2026)5Haslhofer 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Self-Similar Solutions to the Curvature Flow and its Inverse on the 2-dimensional Light Cone

    Fabio Nunes da Silva and Keti TenenblatWider flows

    Topics 3-manifolds, Other curvature flows · Shared terms “light cone”, “dimensional Minkowski”, “Minkowski space”

  2. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows

    Topics Harnack inequalities

  3. Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

    Junyoung ParkWider flows

    Topics Harnack inequalities

  4. The Harnack inequality without convexity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Harnack inequalities

  5. Lectures on curve shortening flow

    Robert HaslhoferWider flows

    Topics Harnack inequalities · Shared terms “differential equation”

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

    Chuanhuan Li, Yi Li, Kairui Xu et al.

    Topics Harnack inequalities

  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

  8. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

    Topics Harnack inequalities

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.