Stephen Lynch
A compact hypersurface in Euclidean space moving by its mean curvature necessarily forms a singularity in finite time. The shape of the hypersurface near a singularity is described by an ancient solution of the flow via a blow-up procedure. Conjecturally, if the original hypersurface has positive mean curvature, then the only blow-up limits that can occur move by self-similarities - they either translate or shrink without changing shape. This has been confirmed for surfaces in three-dimensional space. In higher dimensions, we prove a partial result: all blow-up limits which do not shrink self-similarly are asymptotic to a translating solution. This follows from Hamilton's differential Harnack inequality and a new classification of convex ancient solutions with type I curvature growth
