MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.16 Eternal shrinking helices (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)281263153F94

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).
  2. Stryker–Sun — 2019-08-07 (first arXiv submission).
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(Helical curve-shortening flows) Fix $m\geq 1$ and integers $1\leq k_1<\cdots <k_ m$. Let $r(t)>0$, $t\in \mathbb R$, solve the exact implicit equation

\[ \log r(t)+\frac12\sum _{j=1}^ m r(t)^{2k_ j^2}=\frac m2-t. \]

The choice of constant makes $r(0)=1$. Then

\[ \Gamma _ t(s)= \bigl (r^{k_1^2}\cos (k_1s),r^{k_1^2}\sin (k_1s),\ldots , r^{k_ m^2}\cos (k_ ms),r^{k_ m^2}\sin (k_ ms),s\bigr ), \qquad s\in \mathbb R, \]

is an eternal curve-shortening flow in $\mathbb R^{2m+1}$. In the simplest case $m=1$, $k_1=1$, it is the three-dimensional helix

\[ \Gamma _ t(s)=(r(t)\cos s,r(t)\sin s,s),\qquad \log r(t)+r(t)^2/2=1/2-t. \]

These are exact parametrizations with an implicitly determined time scale, not numerical ODE profiles. The ordinary differential equation is

\[ r'=-\frac{r}{1+\sum _ j k_ j^2r^{2k_ j^2}}. \]

The last coordinate makes every time slice a proper embedding of $\mathbb R$, and the induced metric is $(1+\sum _ jk_ j^2r^{2k_ j^2})ds^2$, so it is complete. It is noncompact with two ends and codimension $2m$, and it spans the stated ambient space. Simultaneous rotations together with translation along the last coordinate give its screw symmetry. Its radius decreases to zero as $t\to +\infty $, giving local smooth convergence to the axis, while $r(t)\to \infty $ as $t\to -\infty $. At any fixed finite time the ends continue winding; they are not asymptotic to the axis as $|s|\to \infty $.

Proposition 2.7 of [SS19] proves

\[ \sup _{t\in \mathbb R}\lambda (\Gamma _ t)=+\infty . \]

This concerns the entire time family and does not say that every individual slice has infinite entropy. In fact, a direct calculation gives the exact fixed-time entropy below. Set $B=\sum _ jr^{2k_ j^2}$ and $D=\sum _ jk_ j^2r^{2k_ j^2}$. Direct Gaussian integration, using $|\Gamma _ t(s)|^2=B+s^2$, gives the exact centered value

\[ F_{0,1}(\Gamma _ t)=\sqrt{1+D}\, e^{-B/4}; \qquad F_{0,1}(\Gamma _ t)=\sqrt{1+r^2}\, e^{-r^2/4} \quad \hbox{when }m=k_1=1. \]

For completeness, the same calculation at arbitrary Gaussian scale gives

\[ \lambda (\Gamma _ t)=\sqrt{1+D(t)}. \]

Indeed, for a center $(a,z_0)\in \mathbb R^{2m}\times \mathbb R$, dropping the nonnegative squared distance in the first $2m$ coordinates bounds $F_{(a,z_0),\tau }$ by $(4\pi \tau )^{-1/2}\sqrt{1+D}\int _{\mathbb R}e^{-(s-z_0)^2/(4\tau )}\, ds=\sqrt{1+D}$. At the origin, $F_{0,\tau }=\sqrt{1+D}e^{-B/(4\tau )}$ tends to this upper bound as $\tau \to \infty $. Thus every finite-time helix has finite entropy, while the time supremum is infinite. This is a direct integration, additional to the cited time-supremum result.

Source: [SS19], equation (2.6), Propositions 2.6–2.7, and Theorem A.2, PDF pp. 6–7 and 9; see also [AAAW13], Section 3, PDF pp. 8–10. The interactive three-dimensional figure uses the simplest exact helix, with a time slider and the displayed range $-5\pi /2\leq s\leq 5\pi /2$. The family is eternal even though a display uses only finitely many turns.

The time slider inverts log r+r²/2=1/2−t. Only −5π/2 ≤ s ≤ 5π/2 is drawn; both infinite ends are omitted. The viewing scale stays fixed.
Exact R3 curve. The time slider inverts log r+r²/2=1/2−t. Only −5π/2 ≤ s ≤ 5π/2 is drawn; both infinite ends are omitted. The viewing scale stays fixed. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Entropy depends on time; infinity is its supremum over the eternal time interval, not a value at any finite time.

R3 helix · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Line ℝ
Embeddedness
Embedded: The last coordinate equals the real parameter, making every finite-time slice a proper embedding.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
2 winding ends
Entropy
Exact formula: λ(Γₜ) = √(1+D(t)), D = Σⱼ kⱼ² r^(2kⱼ²). Finite at each time; supₜ λ = ∞. Direct Gaussian integration.
Gaussian area
Exact formula: F₀,₁(Γₜ) = √(1+D(t)) e^(−B(t)/4), B = Σⱼ r^(2kⱼ²).
Higher-codimension helices · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Line ℝ in ℝ²ᵐ⁺¹, m ≥ 2
Embeddedness
Embedded: The last coordinate equals the real parameter, making every finite-time slice a proper embedding.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
2 winding ends
Entropy
Exact formula: λ(Γₜ) = √(1+D(t)), D = Σⱼ kⱼ² r^(2kⱼ²). Finite at each time; supₜ λ = ∞. Direct Gaussian integration.
Gaussian area
Exact formula: F₀,₁(Γₜ) = √(1+D(t)) e^(−B(t)/4), B = Σⱼ r^(2kⱼ²).

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References on this page

  1. [SS19] Douglas Stryker and Ao Sun. Construction of high codimension ancient mean curvature flows. 2019. Citations use arXiv v2, September 5, 2019; downloaded PDF has 11 pages. Remark 1.3 acknowledges the earlier construction by Altschuler–Altschuler–Angenent–Wu. URL: https://arxiv.org/abs/1908.02688v2, arXiv:1908.02688, doi:10.48550/arXiv.1908.02688.All uses in this project ↩ 1 ↩ 2
  2. [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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