Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes

TL;DR

This paper studies a curvature flow in an unbounded strip where boundary slopes grow with the curve's height, proving convergence to a grim reaper shape despite unbounded boundary conditions.

Contribution

It extends previous convergence results to cases with unbounded boundary slopes, using gradient estimates for symmetric and non-symmetric curves.

Findings

Global gradient estimates are established for symmetric curves.

Uniform interior gradient estimates are derived for general curves.

Curves converge to a grim reaper shape in the specified topology.

Abstract

We consider a curvature flow $V=H$ in the band domain $\Omega :=[-1,1]\times \R$, where, for a graphic curve $\Gamma_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $\Gamma_t$ contacts the two boundaries $\partial_\pm \Omega$ of $\Omega$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $\Gamma_t$ converges to a {\it grim reaper} contacting $\partial_\pm \Omega$ with the same prescribed slopes. In this paper we consider the case where $\Gamma_t$ contacts $\partial_\pm \Omega$ with slopes equaling to $\pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C^{2,1}_{loc} ((-1,1)\times \R)$ topology to the {\it grim reaper} with span $(-1,1)$.