Convergence to the grim reaper for a curvature flow with unbounded boundary slopes

Abstract

We consider a curvature flow \(V=H\) in the band domain \(\Omega :=[-1,1]\times \mathbb {R}\), 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 (Math Ann 295:761–765, 1993) proved that \(\Gamma _t\) converges to a 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 \(C^{2,1}_{loc} ((-1,1)\times \mathbb {R})\) topology to the grim reaper with span \((-1,1)\).