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

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 to 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.

13 pages

References in corpus (3)

• The extension and convergence of mean curvature flow in higher codimension
• Mean curvature flow of star-shaped hypersurfaces
• Stability of spherical collapse under mean curvature flow

Cited by in corpus (3)

• The extension problem of the mean curvature flow (I)
• On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature
• On the extension to mean curvature flow in lower dimension