Self-shrinkers of the mean curvature flow in arbitrary codimension
Self-shrinkers of the mean curvature flow in arbitrary codimension
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DOI：
10.1155/imrn.2005.2983
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发表时间：
2005-07
影响因子：
1
通讯作者：
Knut Smoczyk
中科院分区：
文献类型：
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作者：
Knut Smoczyk
本文研究任意余维平均曲率流的自相似解Mm Rn。Abresch & Langer [AL 86]已经对自相似曲线Γ <$R2进行了完全分类，并且该结果可以同样很好地应用于曲线Γ <$Rn。一个子流形Mm <$Rn称为球面的，如果它包含在一个球面中。显然，平均曲率流的球面自收缩子与球面的极小子流形重合。对于超曲面Mm <$Rm+1，m ≥ 2，Huisken [Hui 90]证明了具有正数量平均曲率的紧致自收缩曲面是球面.我们将证明以下推广：紧自相似解Mm <$Rn，m ≥ 2，是球面的，当且仅当平均曲率向量H非零且主法线ν在法丛中平行。我们还给出了这类完全非紧自收缩子的一个分类。
In this paper 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 & Langer [AL86] 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 hypersurfacesMm ⊂ Rm+1, m ≥ 2, Huisken [Hui90] 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 non-vanishing and the principal normal ν is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.