The harmonic mean curvature flow of nonconvex surfaces in ${\mathbb{R}}^3$

Abstract

We consider a compact, star-shaped, mean convex hypersurface ${\Sigma }^2\subset {\mathbb{R}}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which is star-shaped and mean convex, a smooth solution always exists up to some finite time $T<\infty$ at which the flow shrinks to a point asymptotically spherically.