Self-shrinkers of the mean curvature flow in arbitrary codimension
Differential Geometry
2007-05-23 v1
Abstract
For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution in arbitrary codimension and of dimension greater than one is spherical, i.e. contained in a sphere, if and only if the mean curvature vector \be H\ee is non-vanishing and the principal normal \be\nu\ee is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.
Cite
@article{arxiv.math/0507325,
title = {Self-shrinkers of the mean curvature flow in arbitrary codimension},
author = {Knut Smoczyk},
journal= {arXiv preprint arXiv:math/0507325},
year = {2007}
}
Comments
19 pages, 1 figure