Self-shrinkers of the mean curvature flow in arbitrary codimension

Abstract

We study self-similar solutions Mm ⊂ ℝn of the mean curvature flow in arbitrary codimension. Self-similar curves Γ⊂ℝ2 have been completely classified by Abresch and Langer (1986) and this result can be applied to curves Γ⊂ℝn equally well. A submanifold Mm⊂ℝn is called spherical, if it is contained in a sphere. Obviously, spherical self-shrinkers of the mean curvature flow coincide with minimal submanifolds of the sphere. For hypersurfaces Mm⊂ℝm+1, m ≥ 2, Huisken (1990) showed that compact self-shrinkers with positive scalar mean curvature are spheres. We will prove the following extension: a compact self-similar solution Mm⊂ℝn, m ≥ 2, is spherical, if and only if the mean curvature vector H is nonvanishing and the principal normal ν is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.