TY - JOUR

T1 - Ancient mean curvature flows out of polytopes

AU - Bourni, Theodora

AU - Langford, Mat

AU - Tinaglia, Giuseppe

N1 - Funding Information:
Bourni was supported through grant 707699 of the Simons Foundation.
Funding Information:
Langford acknowledges support from the Australian Research Council through the DECRA fellowship scheme (grant DE200101834).
Publisher Copyright:
© 2018 the Author (s). Published by Kurdistan University of Medical Sciences.

PY - 2022/10/28

Y1 - 2022/10/28

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85142539002&partnerID=8YFLogxK

U2 - 10.2140/gt.2022.26.1849

DO - 10.2140/gt.2022.26.1849

M3 - Article

AN - SCOPUS:85142539002

SN - 1465-3060

VL - 26

SP - 1849

EP - 1905

JO - Geometry and Topology

JF - Geometry and Topology

IS - 4

ER -