MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.20 Rotational shrinking trumpet ends72885E0FF22A

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. Kleene–Moller — 2010-08-10 (first arXiv submission).
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For every positive asymptotic slope, Kleene–Møller construct a rotational shrinking trumpet end in $\mathbb R^{n+1}$, $n\ge 2$, asymptotic to the corresponding round cone. In graph coordinates $r=u(x)>0$ the profile solves $u''=(1+u'^2)(xu'/2-u/2+(n-1)/u)$. The theorem concerns an end with an inner boundary; it must not be presented as a complete boundaryless rotational shrinker. The radial cone is an asymptote, not the trumpet itself. Its smooth rotational sheets are embedded on the stated end, and the geometry is noncompact. We display numerical far-field shooting on a finite interval, with the inner boundary and cutoff explicitly marked. Under the normalization $\mathbf H=-X^\perp /2$, the end evolves as $\sqrt{-t}\, \Sigma $ for $t<0$, including motion of its inner boundary. No boundary condition for a separate fixed-barrier MCF problem is imposed. No global entropy or uniqueness conclusion beyond the cited end problem is recorded.

Sources: [KleeneMoller2014].

Rotational shrinking trumpet ends. A numerical local profile or a labeled orbit-space schematic, with an explicit inner boundary and dashed asymptotic direction. Homothetic time does not remove that boundary.
Numerical ODE end. A numerical local profile or a labeled orbit-space schematic, with an explicit inner boundary and dashed asymptotic direction. Homothetic time does not remove that boundary. 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.

Surface end in ℝ³ · Euclidean space ℝ³ · Not recorded
Ambient space
Euclidean space ℝ³
Topology
Exterior end or orbit-space disk as specified; boundary retained
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Noncompact
Surface genus
Not recorded.
Topological ends
Not recorded for the full parameter manifold.
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.
Higher-dimensional ends · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Not recorded
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Exterior end or orbit-space disk as specified; boundary retained
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Noncompact
Surface genus
Not a surface.
Topological ends
Not recorded for the full parameter manifold.
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. [KleeneMoller2014] Stephen J. Kleene and Niels Martin Moller. Self-shrinkers with a rotational symmetry. Transactions of the American Mathematical Society, 366:3943–3963, 2014. URL: https://arxiv.org/abs/1008.1609, arXiv:1008.1609, doi:10.1090/S0002-9947-2014-05721-8.All uses in this project ↩ 1
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