MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.25 Nonplanar eternal curves from grim reapersAB3265DB1FB6

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–Mramor — 2022-01-06 (first arXiv submission).
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Bourni–Mramor construct a nonplanar eternal curve-shortening flow in $\mathbb R^3$ by joining grim-reaper pieces in distinct planes. The complete embedded slices are ramps: a fixed directional component of the unit tangent has a controlled sign, a property used to prevent singularities and continue the ancient limit forward. The backward picture consists of grim regions in different planes with three parallel straight-line limits; the forward behavior approaches a line in the sense proved in the source. This is not a translating union of grim reapers, nor one of the explicit shrinking helices. The actual solution is obtained by solving curve shortening and taking limits. Its time domain is all of $\mathbb R$; its topology is a noncompact line with two intrinsic ends. A picture of several asymptotic lines is not an end count. The drawing shows the differently oriented grim pieces and a schematic connecting region at successive stages. The source's nonplanarity is intrinsic to the space curve, not an artifact of projection.

Sources: [BourniMramor2023].

Nonplanar eternal curves from grim reapers. Spatial grim-reaper pieces in perpendicular planes, with a separately colored schematic bridge. The three stages illustrate backward grim regions and forward straightening; they are not a numerical eternal solution.
Source-informed schematic. Spatial grim-reaper pieces in perpendicular planes, with a separately colored schematic bridge. The three stages illustrate backward grim regions and forward straightening; they are not a numerical eternal solution. 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.

Nonplanar curve in ℝ³ · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
ℝ
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
Two topological ends.
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. [BourniMramor2023] Theodora Bourni and Alexander Mramor. Nonplanar ancient curve shortening flows in $\mathbb R^3$ from grim reapers. Mathematische Zeitschrift, 2023. URL: https://arxiv.org/abs/2201.02134, arXiv:2201.02134, doi:10.1007/s00209-023-03320-8.All uses in this project ↩ 1
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