Evolution of noncompact hypersurfaces by inverse mean curvature

Abstract

We study the evolution of complete, noncompact, convex hypersurfaces in ${\mathbb{R}}^{n+1}$ by the inverse mean curvature flow. We establish the long-time existence of solutions, and we provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof of a result by Huisken and Ilmanen on compact star-shaped solutions.