  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.
