The global solution of the minimal surface flow and translating surfaces

Abstract

In this paper, we study evolved surfaces over convex planar domains which are evolving by the minimal surface flow $$u_{t}= div\left(\frac{Du}{\sqrt{1+|Du|^2}}\right)-H(x,Du).$$ Here, we specify the angle of contact of the evolved surface to the boundary cylinder. The interesting question is to find translating solitons of the form $u(x,t)=\omega t+w(x)$ where $\omega\in \mathbb R$. Under an angle condition, we can prove the a priori estimate holds true for the translating solitons (i.e., translator), which makes the solitons exist. We can prove for suitable condition on $H(x,p)$ that there is the global solution of the minimal surface flow. Then we show, provided the soliton exists, that the global solutions converge to some translator.