Self-Similar Solutions to the Curvature Flow and its Inverse on the 2-dimensional Light Cone

TL;DR

This paper explores self-similar solutions to curvature and inverse curvature flows on the 2D light cone, characterizing special curves like ellipses and hyperbolas, and establishing their properties and explicit solutions.

Contribution

It establishes a correspondence between solutions of curvature flow and inverse curvature flow on the light cone, characterizes self-similar solutions, and provides explicit examples.

Findings

Ellipses and hyperbolas are the only curves evolving under homotheties.

Ellipses are the only closed self-similar solutions and are ancient solutions.

For each vector, a 2-parameter family of self-similar solutions exists.

Abstract

We show that the solutions to the curvature flow (CF) for curves on the 2-dimensional light cone are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperboles are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant $c$ from being the inner product between its tangent vector field and a fixed vector $v$ of the 3-dimensional Minkowski space. The curve is a soliton solution when $c=0$. We prove that, for each vector $v$ there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Moreover, at each end of such a curve, the curvature is either unbounded or it tends to $0$ or to the constant $c$. Explicitly given soliton solutions are included and some self-similar solutions on the light cone, are visualized.