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

We consider the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> α -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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> α -mean curvature flow of codimension two in the classical sense.