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2022 Ancient mean curvature flows out of polytopes
Theodora Bourni, Mat Langford, Giuseppe Tinaglia
Geom. Topol. 26(4): 1849-1905 (2022). DOI: 10.2140/gt.2022.26.1849

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 n2, 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.

Citation

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Theodora Bourni. Mat Langford. Giuseppe Tinaglia. "Ancient mean curvature flows out of polytopes." Geom. Topol. 26 (4) 1849 - 1905, 2022. https://doi.org/10.2140/gt.2022.26.1849

Information

Received: 10 February 2021; Revised: 20 May 2021; Accepted: 21 June 2021; Published: 2022
First available in Project Euclid: 11 November 2022

Digital Object Identifier: 10.2140/gt.2022.26.1849

Subjects:
Primary: 53E10
Secondary: 52B99

Keywords: ancient solutions , Mean curvature flow , polytopes , translators

Rights: Copyright © 2022 Mathematical Sciences Publishers

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Vol.26 • No. 4 • 2022
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