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Self-shrinkers of the mean curvature flow in arbitrary codimension

Knut Smoczyk
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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 \beν\ee is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.

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Knut Smoczyk. 2005-07-15. Self-shrinkers of the mean curvature flow in arbitrary codimension. https://arxiv.org/abs/math/0507325

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