MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.17 Rotating and translating curve solitons9E45840987F2

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. D. J. Altschuler–S. J. Altschuler–Angenent–Wu — 2012-07-17 (first arXiv submission).
9E45840987F2
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Let $n\geq 3$, $A\ne 0$ be skew-symmetric, and $v\ne 0$ satisfy $Av=0$. The arclength system

\[ C_ s=T,\qquad T_ s=P_ T(AC+v),\qquad |T|=1 \]

has a unique complete solution on $\mathbb R$ for each point and unit tangent. Here $P_ Tw=w-\langle w,T\rangle T$. It generates the eternal curve-shortening flow

\[ X(s,t)=e^{tA}C(s)+tv,\qquad t\in \mathbb R. \]

This is an exact time evolution of an ODE-defined curve in codimension $n-1$. To select genuinely rotating examples, require $P_ T(AC)$ not to vanish identically; the axis line and purely translating curves in $\ker A$ are excluded. A rigidly rotating and translating soliton is different from the shrinking helix in 5.16, whose radius changes with time.

The scalar height $h(s)=\langle v,C(s)\rangle $ satisfies

\[ h_{ss}=-\langle AC+v,T\rangle h_ s+|v|^2. \]

It is either strictly monotone with range $\mathbb R$, or has exactly one global minimum and tends to $+\infty $ at both ends. In the monotone case projection to the translation axis gives an embedding. In the minimum case each branch is a graph over its height range, but this alone does not establish that the two branches are disjoint. All these curves are proper complete immersions with domain $\mathbb R$ and two ends; the height diverges in absolute value at each end. The norm of the projection to $\operatorname {range}A$ tends to either zero or infinity along each end. We do not replace this projection by distance to the translation line when $\dim \ker A>1$.

In $\mathbb R^3$, write $A(x,y,z)=\omega (-y,x,0)$ and $v=be_3$ with $\omega ,b>0$. A family with two upward ends is obtained by choosing $C(0)=(r,0,0)$, $T(0)=(0,1,0)$, $r>0$: $z_ s(0)=0$ and $z_{ss}(0)=b$. This is a concrete initial-value construction, not a closed spatial formula. The illustration uses $r=\omega =b=1$ and shows a finite arclength window. Its time control applies the exact screw motion to the numerical profile. The general higher-dimensional construction allows any nonzero skew map with a nontrivial kernel and any $v\in \ker A\setminus \{ 0\} $.

Source: [AAAW13], Section 4.1 and Lemmas 5.2 and 6.4, PDF pp. 11–12, 14–15, and 20–22. The two height alternatives are read with the proof of Lemma 6.4; its displayed $\pm \infty $ notation does not require opposite signs at the two ends. No global uniqueness with prescribed ends, stability classification, or quantitative Gaussian-area or entropy result is recorded.

Rotating and translating curve solitons. Angular speed and upward translation speed are both 1; C(0) = (1, 0, 0), T(0) = (0, 1, 0), and −24 ≤ s ≤ 24. Both ends ascend from a unique height minimum. A finite numerical window cannot establish global embeddedness. The time control applies exact rotation and translation.
Numerical ODE curve. Angular speed and upward translation speed are both 1; C(0) = (1, 0, 0), T(0) = (0, 1, 0), and −24 ≤ s ≤ 24. Both ends ascend from a unique height minimum. A finite numerical window cannot establish global embeddedness. The time control applies exact rotation and translation. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

A ≠ 0, v ≠ 0, Av = 0, with genuinely nonzero normal rotation. The rotation map must have a kernel. Curves lie in range(A) plus at most a two-dimensional affine kernel component (Lemma 5.2); R4 permits a two-dimensional kernel component.

Monotone-height screw solitons in R3 · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Line R (immersed domain)
Embeddedness
Embedded: Bijective height projection makes the complete proper curve an embedding.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two ends of the parameter manifold R
Entropy
Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area
Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.
Screw solitons with a height minimum in R3 · Euclidean space ℝ³ · Not recorded
Ambient space
Euclidean space ℝ³
Topology
Line R (immersed domain)
Embeddedness
Not recorded: Each branch is a height graph, but disjointness of the two branches is not established by that fact.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two ends of the parameter manifold R
Entropy
Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area
Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.
Monotone-height screw solitons in R4 · Euclidean space ℝ⁴ · Embedded
Ambient space
Euclidean space ℝ⁴
Topology
Line R (immersed domain)
Embeddedness
Embedded: Bijective height projection makes the complete proper curve an embedding.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two ends of the parameter manifold R
Entropy
Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area
Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.
Screw solitons with a height minimum in R4 · Euclidean space ℝ⁴ · Not recorded
Ambient space
Euclidean space ℝ⁴
Topology
Line R (immersed domain)
Embeddedness
Not recorded: Each branch is a height graph, but disjointness of the two branches is not established by that fact.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two ends of the parameter manifold R
Entropy
Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area
Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.
Monotone-height screw solitons in R^n, n ≥ 5 · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Line R (immersed domain)
Embeddedness
Embedded: Bijective height projection makes the complete proper curve an embedding.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two ends of the parameter manifold R
Entropy
Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area
Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.
Screw solitons with a height minimum in R^n, n ≥ 5 · Euclidean dimension ≥ 5 · Not recorded
Ambient space
Euclidean dimension ≥ 5
Topology
Line R (immersed domain)
Embeddedness
Not recorded: Each branch is a height graph, but disjointness of the two branches is not established by that fact.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two ends of the parameter manifold R
Entropy
Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area
Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

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References on this page

  1. [AAAW13] Dylan J. Altschuler, Steven J. Altschuler, Sigurd B. Angenent, and Lani F. Wu. The zoo of solitons for curve shortening in ℝⁿ. Nonlinearity, 26(5):1189–1226, 2013. Citations use arXiv v1, July 17, 2012, 39 pages; Sections 1–8 reviewed for the curve-soliton expansion; downloaded PDF has a later typesetting date on its title page. URL: https://arxiv.org/abs/1207.4051v1, doi:10.1088/0951-7715/26/5/1189.All uses in this project ↩ 1
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