MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.30 Ancient dense torus curves with irrational frequenciesE22AF7044670

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).
E22AF7044670
3E693532AA77

Choose $m\geq 2$, amplitudes $a_ j>0$, and positive frequencies $\omega _ j$ which are linearly independent over $\mathbb Q$. For $\theta ,\tau \in \mathbb R$, set

\[ C(\theta ,\tau )=\bigl (a_ j e^{-\omega _ j^2\tau }\cos (\omega _ j\theta ), a_ j e^{-\omega _ j^2\tau }\sin (\omega _ j\theta )\bigr )_{j=1}^ m, \qquad t(\tau )=-\frac12\sum _{j=1}^ m a_ j^2e^{-2\omega _ j^2\tau }. \]

The strictly increasing map $\tau \mapsto t$ maps $\mathbb R$ onto $(-\infty ,0)$. These formulas therefore define an ancient curve-shortening flow in $\mathbb R^{2m}$, of dimension one and codimension $2m-1$. They are explicit in the auxiliary time $\tau $; physical time generally requires inversion of the displayed scalar relation. Indeed, with

\[ D(\tau )=\sum _ j\omega _ j^2a_ j^2e^{-2\omega _ j^2\tau }, \qquad \frac{dt}{d\tau }=D, \]

one has $|C_\theta |^2=D$, $C_\tau =C_{\theta \theta }$, and $C_ t=C_{ss}$. This direct differentiation fixes the coefficient in the time relation.

At each finite time the curve is an injective immersion of $\mathbb R$, dense in the product of the $m$ coordinate circles. It is not an embedding: recurrence gives sequences escaping in the parameter whose images converge to a point of the curve. It is also not proper. Its constant positive speed makes its induced metric complete. Its parameter manifold is noncompact with two intrinsic ends, although its image is bounded and neither end escapes every compact set. The closure is a torus; the evolving object is a curve, not that torus as a surface. The flow has simultaneous rotational symmetry and is not homothetically self-similar because distinct coordinate radii have changing ratios.

The exact radius and curvature identities are

\[ |C|^2=-2t,\qquad |C_{ss}|^2=\frac{\sum _ j\omega _ j^4 a_ j^2e^{-2\omega _ j^2\tau }}{D(\tau )^2}. \]

Thus curvature is constant along each time slice. After dividing by $\sqrt{-2t}$, the largest-frequency coordinate pair dominates as $t\to -\infty $, and the smallest-frequency pair dominates as $t\uparrow 0$. The locally parametrized limits are great circles traced by an unbounded parameter; no finite-multiplicity convergence is asserted for the full recurrent curve.

At a fixed time, the infinite length lies in a bounded set. Consequently every positive-scale Gaussian length integral is infinite, and both $F_{0,1}$ and entropy are $+\infty $. This is a direct consequence of the parametrization, with length counted on the immersed domain. No finite Gaussian-area or locally finite length assertion is intended.

For the illustrated $\mathbb R^4$ case take $a_1=a_2=1$ and $(\omega _1,\omega _2)=(1,\sqrt2)$. An explicit physical-time formula is obtained by setting

\[ u(t)=\frac{\sqrt{1-8t}-1}{2},\qquad C(\theta ,t)=\bigl (\sqrt u\cos \theta ,\sqrt u\sin \theta , u\cos (\sqrt2\theta ),u\sin (\sqrt2\theta )\bigr ),\qquad t<0. \]

Source: [AAAW13], Section 3, equations (11)–(14), PDF pp. 8–10. The irrational-frequency case is distinct from the compact rational-frequency family in 5.29. The figure samples a finite parameter interval and projects four-dimensional geometry; apparent crossings in this projection do not establish self-intersections in $\mathbb R^4$. No stability classification is recorded.

Ancient dense torus curves with irrational frequencies. Frequencies (1, √2), amplitudes one, and physical time t = −1. The finite interval −12π ≤ θ ≤ 12π samples an injectively immersed curve dense in a torus; the complete curve is nonproper. Time and fourth-coordinate rotation controls preserve the exact formula. Projected crossings are not self-intersections in R4.
Exact R4 curve, projected to R3. Frequencies (1, √2), amplitudes one, and physical time t = −1. The finite interval −12π ≤ θ ≤ 12π samples an injectively immersed curve dense in a torus; the complete curve is nonproper. Time and fourth-coordinate rotation controls preserve the exact formula. Projected crossings are not self-intersections in R4. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Only even ambient dimensions 2m ≥ 4 are recorded. The closure is an m-torus, while the flow has intrinsic dimension one. Its infinite Gaussian length counts the immersed domain.

Two rationally independent frequencies in R4 · Euclidean space ℝ⁴ · Not embedded
Ambient space
Euclidean space ℝ⁴
Topology
Line R (immersed domain)
Embeddedness
Not embedded: Injective but recurrent: the inverse onto its image is not continuous. This is a nonproper immersion, not an embedding.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two intrinsic ends; both remain in a bounded torus
Entropy
Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area
Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).
m rationally independent frequencies in R^(2m), m ≥ 3 · Euclidean dimension ≥ 5 · Not embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Line R (immersed domain)
Embeddedness
Not embedded: Injective dense torus orbit with noncontinuous inverse and nonproper image parametrization.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Two intrinsic ends; neither escapes to infinity
Entropy
Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area
Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).

Compare examples by these properties →

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
Cite this page

Comments

Comments are not enabled yet.