Polygonal Pancakes (Joint seminar with UCSD)

Confirming the heuristic described by Huisken and Sinestrari, we show that the Grim reaper gives rise to a great diversity of convex ancient and translating solutions to mean curvature flow, through the evolution of families of Grim hyperplanes in suitable configurations. We construct, in all dimensions $n\ge 2$, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. The latter are counterexamples to a conjecture of White. We also provide a detailed asymptotic analysis of convex ancient solutions in slab regions in general. Roughly speaking, we show that they decompose ``backwards in time'' into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. An analogous decomposition holds ``forwards in time'' for eternal solutions. One consequence is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across the mid-plane of their slab. This is joint work with Mat Langford and Giuseppe Tinaglia.