Mean curvature flow with free boundary on smooth hypersurfaces

The classical mean curvature flow of hypersurfaces with boundary satisfying a Neumann condition on an arbitrary, fixed, smooth hypersurface in Euclidean space is examined. In particular, the problem of singularity formation on the free-boundary and the classification of the limiting behaviour thereof is focused on. A monotonicity formula is developed and used to show that any smooth blow up centred about a boundary point is self-similar, with smoothness of the blow up being shown to necessarily follow in the case of Type I singularities. This leads to a classification of boundary singularities for mean convex evolving hypersurfaces.