  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.

Differential Geometry