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Ancient mean curvature flows out of polytopes

### Theodora Bourni, Mat Langford and Giuseppe Tinaglia

Geometry & Topology 26 (2022) 1849–1905
##### Abstract

We develop a theory of convex ancient mean curvature flow in slab regions, with Grim hyperplanes playing a role analogous to that of half-spaces in the theory of convex bodies.

We first construct a large new class of examples. These solutions emerge from circumscribed polytopes at time minus infinity and decompose into corresponding configurations of “asymptotic translators”. This confirms a well-known conjecture attributed to Hamilton; see also Huisken and Sinestrari (2015). We construct examples in all dimensions $n\ge 2$, which include both compact and noncompact examples, and both symmetric and asymmetric examples, as well as a large family of eternal examples that do not evolve by translation. The latter resolve a conjecture of White (2003) in the negative.

We also obtain a partial classification of convex ancient solutions in slab regions via a detailed analysis of their asymptotics. Roughly speaking, we show that such solutions decompose at time minus infinity into a canonical configuration of Grim hyperplanes. An analogous decomposition holds at time plus infinity for eternal solutions. There are many further consequences of this analysis. One is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across the midplane of their slab.

##### Keywords
polytopes, mean curvature flow, ancient solutions, translators
Primary: 53E10
Secondary: 52B99