MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.25 Drugan–Kleene immersed shrinking familiesF2D20223E5E5

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. Drugan–Kleene — 2013-06-10 (first arXiv submission).
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For every $n\geq 2$, Drugan–Kleene construct infinitely many complete rotational immersed self-shrinkers in $\mathbb R^{n+1}$ of each topology $S^ n$, $\mathbb R^ n$, $\mathbb R\times S^{n-1}$ and $S^1\times S^{n-1}$. This expands the single sphere construction in Example 2.24; it does not identify the torus examples with Angenent's embedded torus. The arclength meridian $(x,r)$ satisfies

\[ x'=\cos \alpha ,\quad r'=\sin \alpha ,\quad \alpha '=\frac{x}{2}\sin \alpha +\left(\frac{n-1}{r}-\frac r2\right)\cos \alpha . \]

A smooth axis endpoint must meet the axis orthogonally. Closed meridians generate tori; the other endpoint conditions distinguish spheres, planes and cylinders. The existence proof uses shooting and limiting geodesics; arbitrary initial data do not produce one of these complete examples. The flow is $M_ t=\sqrt{-t}\, \Sigma $, $-\infty <t<0$. Nontrivial rotational spheres, planes and cylinders in this construction are not embedded, by the rotational classification quoted in the source. No blanket embedding or uniqueness assertion is made for the torus branch.

Sources: [DruganKleene2017].

Drugan--Kleene immersed shrinking families. Meridian construction diagram: closed loops and returning/escaping branches distinguish the four topologies. The three panels apply the exact shrinker time scaling to schematic profiles, not to a numerically certified complete Drugan–Kleene shrinker.
Source-informed schematic. Meridian construction diagram: closed loops and returning/escaping branches distinguish the four topologies. The three panels apply the exact shrinker time scaling to schematic profiles, not to a numerically certified complete Drugan–Kleene shrinker. 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 cases, n = 2 · Euclidean space ℝ³ · Depends on the construction
Ambient space
Euclidean space ℝ³
Topology
S², ℝ², ℝ × S¹, or S¹ × S¹
Embeddedness
Depends on the construction: The nontrivial sphere, plane and cylinder branches are nonembedded. Torus embeddedness is not asserted uniformly.
Compactness
Not specified for this subfamily
Surface genus
Depends on the construction.
Topological ends
Depends on the input; asymptotic wings are not automatically 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.
Hypersurface cases, n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Depends on the construction
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Sⁿ, ℝⁿ, ℝ × Sⁿ⁻¹, or S¹ × Sⁿ⁻¹
Embeddedness
Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness
Not specified for this subfamily
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
Depends on the input; asymptotic wings are not automatically 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. [DruganKleene2017] Gregory Drugan and Stephen J. Kleene. Immersed self-shrinkers. Transactions of the American Mathematical Society, 369:7213–7250, 2017. URL: https://arxiv.org/abs/1306.2383, arXiv:1306.2383, doi:10.1090/tran/6907.All uses in this project ↩ 1
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