Self-shrinkers of the mean curvature flow in arbitrary codimension

By Knut Smoczyk. 2005

106 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1039b6af37d5999a20672c0b4249eb3515372fb2

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In Applied Mathematics

1. Qing Cheng
2. Yejuan Peng
3. Y. Xin
4. Haizhong Li
5. Huaihuo Cao
6. Qi Ding
7. Ovidiu Munteanu
8. Qiang Chen

In Geometry and Topology

1. Qing Cheng
2. Yejuan Peng
3. Y. Xin
4. Haizhong Li
5. Huaihuo Cao
6. Qi Ding
7. Ovidiu Munteanu
8. Qiang Chen

In Mathematical Physics

1. Qing Cheng
2. Yejuan Peng
3. Y. Xin
4. Huaihuo Cao
5. Haizhong Li
6. Qi Ding
7. Hongcang Yang

Citations in the corpus

• The rigidity theorems of self-shrinkers (2011, methodology): “For arbitrary submanifolds in Euclidean space, a Simons type formula was also derived (see [21], [22], for example).”
• Complete self-shrinkers of the mean curvature flow (2012, background): “Smoczyk in [22] proved that let M be a complete self-shrinker with H 6= 0 and with parallel principal normal vector ν = H/|H| in the normal bundle, if M has uniformly bounded geometry, then M must be Γ×Rn−1 or M̃ r×Rn−r.”
• A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension (2011, result): “Smoczyk [15] proved the following two results: (i) For any n-dimensional compact self-shrinker M in R satisfying (1.”
• Volume growth, eigenvalue and compactness for self-shrinkers (2011, unknown): “For higher codimension, there is a few study, see [15] for example.”