Self-expanders of the mean curvature flow

Abstract

We study self-expanding solutions $M^m\subset\mathbb{R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function $|A|^2/|H|^2$ attains a local maximum, where $A$ denotes the second fundamental form and $H$ the mean curvature vector of $M$. If the pricipal normal $\xi=H/|H|$ is parallel in the normal bundle, then a similar result holds in higher codimension for the function $|A^\xi|^2/|H|^2$, where $A^\xi$ is the second fundamental form with respect to $\xi$. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension $2$ any mean convex self-expander that is asymptotic to a cone must be strictly convex.