Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone

Abstract

We investigate the evolution of hypersurfaces with perpendicular Neumann boundary condition under mean curvature type flow, where the boundary manifold is a convex cone. We find that the volume enclosed by the cone and the evolving hypersurface is invariant. By maximal principle, we prove that the solutions of this flow exist for all time and converge to some part of a sphere exponentially as $t$ tends to infinity.