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

Authors
Type
Preprint
Publication Date
Sep 03, 2008
Submission Date
Jun 10, 2008
Source
arXiv
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.