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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 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
Mathematical Subject Classification
Primary: 53E10
Secondary: 52B99
References
Publication
Received: 10 February 2021
Revised: 20 May 2021
Accepted: 21 June 2021
Published: 28 October 2022
Proposed: Tobias H Colding
Seconded: Bruce Kleiner, Dmitri Burago
Authors
Theodora Bourni
Department of Mathematics
University of Tennessee Knoxville
Knoxville, TN
United States
http://www.math.utk.edu/~bourni/
Mat Langford
School of Mathematical and Physical Sciences
The University of Newcastle
Newcastle, NSW
Australia
http://www.math.utk.edu/~langford/
Giuseppe Tinaglia
Department of Mathematics
King’s College London
London
United Kingdom
https://nms.kcl.ac.uk/giuseppe.tinaglia/