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

TL;DR

This paper studies curve flows on the light-cone in Minkowski space, deriving inequalities and classifying space-periodic solitons expressed via elliptic functions, revealing a family of transcendental solutions with specific rotation indices.

Contribution

It provides a detailed classification of space-periodic solitons for a third-order curvature flow on the light-cone, including explicit elliptic function solutions and their properties.

Findings

Derived the Harnack inequality for the heat flow.

Classified space-periodic solitons using Jacobi elliptic functions.

Identified a family of transcendental closed soliton solutions characterized by rotation index p/q.

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.