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 our main results is, that complete convex hypersurfaces are products curves and flat subspaces, if only function | A 2 /| H attains a local maximum, where denotes second fundamental form vector M . If principal normal ξ = is parallel in bundle, then similar result holds higher codimension for , with respect to As corollary we obtain self-expanders attain strictly positive scalar curvature, they smoothly asymptotic cones non-negative curvature. In particular, dimension any self-expander cone must be convex.