Blow-up of the mean curvature at the first singular time of the mean curvature flow

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Abstract

It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal than 7. We show that the mean curvature blows up at the singularities of the mean curvature flow starting from an immersed closed hypersurface with small \(L^2\)-norm of the traceless second fundamental form (observe that the initial hypersurface is not necessarily convex). As a consequence of the proof of this result we also obtain the dynamic stability of a sphere along the mean curvature flow with respect to the \(L^2\)-norm.

Mathematics Subject Classification

53C44 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of California, Santa CruzSanta CruzUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

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