MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.20 Convex slab translators with polyhedral asymptotics2F596CC616F1

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. Bourni–Langford–Tinaglia — 2020-06-29 (first arXiv submission).
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Bourni–Langford–Tinaglia construct convex slab translators from an unbounded regular circumscribed polytope $P\subset \mathbb R^ n$ that does not split a line, and separately from circumscribed unbounded simplices. Here circumscribed means that the defining facets are tangent to the unit sphere with the orientations in the source; regularity is the source's symmetry condition on the polytope, not arbitrary polyhedral data. The resulting hypersurface lies in $(-\pi /2,\pi /2)\times \mathbb R^ n$, lies in no smaller slab, and its translating evolution sweeps out that slab. Its blow-down under the linear ancient-time scale (the source's “squash-down”) is $P$. The regular construction is locally uniformly convex, reflection symmetric in the slab midplane, and inherits the symmetries of $P$. The evolution is an exact translation of the constructed hypersurface for all real times; the speed and direction are determined by the polyhedral data. There is no closed height formula or general uniqueness claim here. The diagram illustrates a higher-dimensional polyhedral section, not the curvature or topology of a surface in $\mathbb R^3$. Lower-dimensional overlapping slab examples are cross-referenced to Example 3.9.

Sources: [BourniLangfordTinaglia2022].

Convex slab translators with polyhedral asymptotics. Cross-section of the prescribed circumscribed polyhedral asymptotics. Moving copies illustrate translation; the smooth translator and its speed are not numerically solved.
Source-informed schematic. Cross-section of the prescribed circumscribed polyhedral asymptotics. Moving copies illustrate translation; the smooth translator and its speed are not numerically solved. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Higher-dimensional polyhedral constructions, n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Complete convex graphical hypersurface; polyhedral asymptotic data
Embeddedness
Embedded: The cited construction gives an embedding for this case.
Compactness
Noncompact
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
One topological end.
Entropy
Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area
Not recorded: No quantitative conclusion is recorded from the cited construction.

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References on this page

  1. [BourniLangfordTinaglia2022] Theodora Bourni, Mat Langford, and Giuseppe Tinaglia. The atomic structure of ancient grain boundaries. Geometry & Topology, 26:1849–1905, 2022. URL: https://arxiv.org/abs/2006.16338, arXiv:2006.16338, doi:10.2140/gt.2022.26.1849.All uses in this project ↩ 1
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