Existence of Mean Curvature Flow Singularities with Bounded Mean Curvature

Abstract

In [Vel94], Velazquez 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 $N \ge 8$, a nontrivial subset of these solutions has uniformly bounded mean curvature.