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• W3040285679 abstract "Democritus and early atomists held that the material cause of all things that exist is coming together of atoms and void. Atoms are eternal and have many different shapes, and they can cluster together to create things that are perceivable. Differences in shape, arrangement, and position of atoms produce different phenomena. Like atoms of Democritus, Grim Reaper solution to curve shortening flow is eternal and indivisible -- it does not split off a line, and is itself its only translator. Confirming heuristic described by Huisken and Sinestrari [J. Differential Geom. 101, 2 (2015), 267-287], we show that it gives rise to a great diversity of convex ancient and translating solutions to mean curvature flow, through evolution of families of Grim hyperplanes in suitable configurations. We construct, in all dimensions $nge 2$, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. The latter resolve a conjecture of White [J. Amer. Math. Soc. 16, 1 (2003), 123-138]. We also provide a detailed asymptotic analysis of convex ancient solutions in slab regions in general. Roughly speaking, we show that they decompose backwards in time into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. An analogous decomposition holds forwards in time for eternal solutions. One consequence is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across mid-plane of their slab." @default.
• W3040285679 created "2020-07-10" @default.
• W3040285679 creator A5000849080 @default.
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• W3040285679 date "2020-06-29" @default.
• W3040285679 modified "2023-09-27" @default.
• W3040285679 title "The atomic structure of ancient grain boundaries" @default.
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