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Ancient mean curvature
flows out of polytopes
Theodora Bourni, Mat Langford and Giuseppe Tinaglia |
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Geometry & Topology 26 (2022) 1849–1905
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Abstract |
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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 , 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
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Mathematical Subject Classification
Primary: 53E10
Secondary: 52B99
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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
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Authors |
| Theodora Bourni | |
| Department of Mathematics University of Tennessee Knoxville Knoxville, TN United States |
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| http://www.math.utk.edu/~bourni/ | |
| Mat Langford | |
| School of Mathematical and Physical
Sciences The University of Newcastle Newcastle, NSW Australia |
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| http://www.math.utk.edu/~langford/ | |
| Giuseppe Tinaglia | |
| Department of Mathematics King’s College London London United Kingdom |
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| https://nms.kcl.ac.uk/giuseppe.tinaglia/ | |
