Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers

By Nam Q. Le, N. Šešum. 2010

95 citations recorded by the source corpus.

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in \cite{St} coincide for any Type I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex Type I Mean curvature flow. We also establish a gap theorem for self-shrinkers.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f68a7766c201f0eb31f359902aa1bc28f6a7de2e

How researchers see this paper

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In Applied Mathematics

1. Qing Cheng
2. Haizhong Li
3. G. Wei
4. Huaihuo Cao
5. Knut Smoczyk
6. Ovidiu Munteanu
7. Qiang Chen
8. Jiaping Wang

In Geometry and Topology

1. Qing Cheng
2. Haizhong Li
3. G. Wei
4. Huaihuo Cao
5. Knut Smoczyk
6. Ovidiu Munteanu
7. Qiang Chen
8. Jiaping Wang

In Mathematical Physics

1. Qing Cheng
2. Haizhong Li
3. Huaihuo Cao
4. G. Wei
5. Knut Smoczyk
6. Hongcang Yang
7. Yejuan Peng
8. Y. Xin

Citations in the corpus

• A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension (2011, background): “7 in [11]) for self-shrinkers of codimension 1: if a hypersurface Mn ⊂ Rn+1 is a smooth complete embedded self-shrinker without boundary and with polynomial volume growth, and satisfies |A|2 < 1, then Mn is a hyperplane.”
• Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, methodology): “As was recently shown by Le and Sesum [58] this does not happen in the case of Type I singularities of hypersurfaces in RmC1 and all notions of singular sets defined in [81] coincide.”
• A GAP THEOREM OF SELF-SHRINKERS (2012, background): “Motivated by the above results of Le and Sesum, Cao and Li, Ding and Xin, we consider the second gap for the squared norm of the second fundamental form and prove the following classification theorem for self-shrinkers: Theorem 1.1.”
• The rigidity theorems of self-shrinkers (2011, unknown): “The first gap of the squared norm of the second fundamental form for selfshrinkers was obtained by Cao-Li [2] (which generalized the codimension 1 case in [15]).”