The ancient pancake

In Euclidean space of dimension at least two there exist ancient solutions of mean curvature flow with compact, convex time slices which lie at all times in a given slab region (the region between two parallel hyperplanes). Up to rigid motions, time translations and parabolic scaling, there is a unique such solution in the class of rotationally symmetric solutions. It turns out that this solution is also reflection symmetric across the 'midplane' of the slab. In the Euclidean plane, it is the well-known Angenent oval solution, whose diameter grows like ‐t+log 2+o(1) as ‐t goes to infinity. In higher dimensions, its diameter grows like ‐t+(n‐1)log(‐t)+cn+o(1) as ‐t goes to infinity, where n is the dimension of the solution and cn is a constant which only depends on n.

The following .gif files depict the evolution of the two-dimensional ancient pancake.

The ancient pancake [top view] [side view]