Generalized Elastic Translating Solitons

TL;DR

This paper characterizes translating solitons for curvature flows in the plane as generalized elastic curves, providing new variational insights especially for the curve shortening flow and the grim reaper solution.

Contribution

It introduces a novel variational characterization of translating solitons as generalized elastic curves, extending understanding of curvature flow solutions.

Findings

Translating solitons are critical points of specific curvature-dependent functionals.

The grim reaper curve is characterized variationally within this framework.

The study applies to flows by powers of curvature, including the classical curve shortening flow.

Abstract

We study translating soliton solutions to the flow by powers of the curvature of curves in the plane. We characterize these solitons as critical curves for functionals depending on the curvature. More precisely, translating solitons to the flow by powers of the curvature are shown to be generalized elastic curves. In particular, focusing on the curve shortening flow, we deduce a new variational characterization of the grim reaper curve.