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

Yun Yang

TL;DR

This work studies curve flows on the light-cone $LC^*$ in ${\mathbb E}_1^3$, deriving a Li–Yau type Harnack inequality for the heat flow and analyzing a non-stretching third-order curvature flow that reduces to a KdV-type equation. By exploiting Killing fields on $LC^*$, the authors classify space-periodic solitons of the third-order flow and construct explicit analytic soliton solutions in terms of Jacobi elliptic functions, including a closed-family $x_{p,q}$ with $p/q\in(\sqrt{2/3},1)$. The soliton solutions are organized by rotation axis (time-like, light-like, space-like), with detailed formulas for the curvature, rotation angle per period, and period lengths expressed via elliptic integrals; the study also yields an analytic solution to an associated second-order nonlinear ODE. Overall, the paper advances explicit soliton constructions and integrable-structure insights for curve flows in Lorentzian geometry on light-cones, with connections to centro-affine geometry and elliptic-function methods.

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 $\mathrm{x}_{p,q}$, which are characterized by a rotation index $p$ and close after $q$ periods of their curvature functions. The ratio $p/q$ satisfies $p/q \in (\sqrt{2/3}, 1)$, where $p$ and $q$ are relatively prime positive integers. Guided by the classification process, we obtain the analytic solutions to a second-order nonlinear ordinary differential equation.

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

TL;DR

This work studies curve flows on the light-cone in , deriving a Li–Yau type Harnack inequality for the heat flow and analyzing a non-stretching third-order curvature flow that reduces to a KdV-type equation. By exploiting Killing fields on , the authors classify space-periodic solitons of the third-order flow and construct explicit analytic soliton solutions in terms of Jacobi elliptic functions, including a closed-family with . The soliton solutions are organized by rotation axis (time-like, light-like, space-like), with detailed formulas for the curvature, rotation angle per period, and period lengths expressed via elliptic integrals; the study also yields an analytic solution to an associated second-order nonlinear ODE. Overall, the paper advances explicit soliton constructions and integrable-structure insights for curve flows in Lorentzian geometry on light-cones, with connections to centro-affine geometry and elliptic-function methods.

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.

Paper Structure

This paper contains 12 sections, 17 theorems, 130 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Frenet formulas in Minkowski space
  4. The Frenet formulas on light-cone
  5. The evolutions of curves on light-cone
  6. Harnack inequality on light-cone
  7. A third order curvature flow
  8. The soliton solutions for the third order curvature flow
  9. Solitons rotating around a time-like axis
  10. Solitons rotating around a light-like axis
  11. Solitons rotating around a space-like axis
  12. Proof of Theorem \ref{['thm-ode']}

Key Result

Theorem 1.2

For an immersed solution to the curve flow curve-evo-light on the light-cone $LC^*$ with $k>0$ and defined on the time interval $[t_0,T)$, we have the following Harnack inequality

Figures (4)

  • Figure 1: The closed solitons on light-cone.
  • Figure 2: The non-closed periodic solitons on light-cone.
  • Figure 3: The comparison of different method to calculate $\Lambda^{\Theta}$.
  • Figure 4: Monotonicity & asymptotics at $s=T$ with varying $\lambda$.

Theorems & Definitions (40)

  • Example 1.1
  • Theorem 1.2: Harnack Inequality
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 30 more