On the singular set of mean curvature flows with Neumann free boundary conditions

We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of the singular set is zero. In fact, we consider two types of interaction between the support and flowing surfaces. In the case of weaker interaction, we need make no further assumptions than in the case without boundary to achieve our result. In the case of stronger interaction, we need only make the additional assumption that $H_{\Sigma}>0$, that is, that the support surface be mean convex. We go on, in this case, to show that the result is not, in general, true without the mean convexity assumption.