Existence of mean curvature flow singularities with bounded mean curvature

Abstract

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 ${\mathbb{R}}^N$ in every dimension $N\ge 8$. 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 $N\ge 8$, infinitely many of these solutions have uniformly bounded mean curvature.