α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets

Abstract

We consider the α\alpha -mean curvature flow for convex graphs in Euclidean space. Given a smooth, complete, strictly convex, non-compact initial hypersurface over a strictly convex projected domain, we derive uniform curvature bounds, which are independent of the height of a graph, to give C2{C}^{2}-estimates for convex graphs. Consequently, these height-independent estimates imply that all the derivatives for level sets converge uniformly. Furthermore, with these estimates on level sets, the boundary of the domain of a graph, which demonstrates the behavior of level sets as the height tends to infinity, is shown to be a smooth solution for the α\alpha -mean curvature flow of codimension two in the classical sense.

Keywords

• mean curvature flow
• graph
• 53e10
• 53c21