Self-Expanders of the Mean Curvature Flow

Abstract

We study self-expanding solutions $M^{m}\subset \mathbb {R}^{n}$ M m ⊂ 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 principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. 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.

On $n$-dimensional complete self-similar solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature

“…Moreover, in some situations uniqueness of self-expanders is important for the construction of mean curvature flows starting from certain singular configurations [BM]. The interested reader can consult the recent article [Sm2] by Smoczyk and references therein for more recent progress and a historical note on self-expanders.…”