A variant of inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone

Abstract

Given a smooth convex cone in the Euclidean \((n+1)\)-space (\(n\geq 2\)), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a variant of inverse mean curvature flow, then, by using the convexity of the cone in the derivation of the gradient and Hölder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.