Existence of mean curvature flow singularities with bounded mean curvature

In “Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow,” Velázquez constructed a countable collection of mean curvature flow solutions in RN in every dimension N≥8. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension N≥8, infinitely many of these solutions have uniformly bounded mean curvature.