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

TL;DR

This paper proves that the mean curvature must blow up at the first singularity of the mean curvature flow for certain initial hypersurfaces, supporting the conjecture in dimensions up to 7 and demonstrating sphere stability.

Contribution

It establishes the blow-up of mean curvature at singularities for hypersurfaces with small L^2-norm of traceless second fundamental form, extending understanding beyond convex cases.

Findings

Mean curvature blows up at singularities for specific initial conditions.

Sphere stability along mean curvature flow in L^2-norm.

Supports the conjecture for dimensions ≤ 7.

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.