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

TL;DR

This paper studies the evolution of complete, mean convex hypersurfaces in Euclidean space under a flow driven by a positive power of mean curvature, proving long-time existence through approximation methods.

Contribution

It establishes the long-time existence of the $H^{ ext{ extalpha}}$-flow for complete, mean convex graphs, extending previous results to unbounded hypersurfaces.

Findings

Long-time existence of the flow is proven.

Approximation by bounded problems is used.

Results apply to complete, mean convex graphs.

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.