$H^{\alpha }$-flow of mean convex, complete graphical hypersurfaces

Abstract

We consider the evolution of hypersurfaces in ${\mathbb{R}}^{n+1}$ with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature), complete, and given as the graph of a function. Long-time existence of the $H^{\alpha }$-flow is established by means of approximation by bounded problems.