MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.6 Eternal curves from chains of grim reapers (Angenent–You, 2018; Choi–Seo–Su–Zhao, 2026)321CCF32BF3F

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. Angenent–You — 2018-03-04 (first arXiv submission).
  2. Choi–Seo–Su–Zhao — 2026-03-10 (first arXiv submission).
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Choose monotonically ordered heights $a_0<a_1<\cdots <a_ m$ and horizontal shifts $C_1,\ldots ,C_ m$. Angenent–You construct embedded eternal planar curve-shortening flows which for sufficiently negative time are assembled asymptotically from alternating grim reapers on consecutive strips. To record a definite nontrivial subfamily, take $m=3$. With $v_ k=\pi /(a_ k-a_{k-1})$, the separate asymptotic models are

\[ x=C_ k+(-1)^ k\left[-v_ k^{-1}\log \sin (v_ k(y-a_{k-1}))+v_ kt\right], \qquad a_{k-1}<y<a_ k. \]

This formula describes the model translators, not the glued eternal solution.

The actual flow is a graph $x=u(y,t)$ on $a_0<y<a_3$, exists for every $t\in \mathbb R$, and is complete, properly embedded and noncompact, with topology $\mathbb R$ and two ends asymptotic to the outer horizontal lines. The two-end and properness statements follow from this graphical description and the outer asymptotes. Curvature changes sign in the nontrivial chain; total absolute curvature is uniformly bounded. There is no closed formula for $u$; the classification and flow entropy are qualified below.

Source: [AY18], equation (3), Section 3.1 and Lemmas 3.8–3.10, pp. 2 and 8–13. The paper also constructs immersed ancient chains; those are not being labeled embedded by this selected case. The illustration uses $a_ k=k\pi $ and $C_ k=0$, so that the three model tips have horizontal velocities $-1,+1,-1$. It shows a numerical finite-start approximation: smoothly joined reapers at $t=-10$ are evolved by curve shortening, and slices from $t=-6$ to $t=9$ are displayed at the same spatial scale. The outer ends are truncated far outside the viewing window. The joins follow the construction in Section 3.2 and Figure 10, using the orientation of equation (3) and Figure 9. This computation is not a closed formula or a certified approximation of the eternal limit; the later shapes are numerical observations, not an additional asymptotic theorem. In particular, $t=0$ is not an extinction time.

The classification in [ChoiSeoSuZhao2026], Theorem 1.1, identifies graphical ancient trombones as the nonconvex embedded planar ancient flows with finite flow entropy, alongside the line, shrinking circle, paperclip and grim reaper. Here flow entropy means $\operatorname {Ent}[M]=\sup _ t\lambda (M_ t)<\infty $, not merely finite entropy of each slice. Corollary 1.5 records flow entropy $m+1$ for the $m$-reaper family; thus the selected $m=3$ case has flow entropy four. This does not identify the entropy or centered Gaussian area of an individual slice with four.

Six numerical time slices of one planar curve with three alternating grim-reaper turns, evolving from separated turns to a broader profile. Both outer ends continue to the left along y = 0 and y = 3π. Finite-start numerical approximation.
Numerical curve-shortening flow. Angenent–You, equation (3), §3.1–3.3 and Figure 9. Equal strips aₖ = kπ and shifts Cₖ = 0. Play the evolution or compare six times from t = −6 to 9 at a fixed scale. The model overlay shows exact separate grim reapers, with velocities −1, +1, −1. The blue curve is computed from smoothly joined data at t = −10, with distant truncated ends; it is not a closed formula or a certified approximation of the eternal limit. Zero is an ordinary time, not an extinction time. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
Monotone three-reaper chain · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Line ℝ
Embeddedness
Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
2
Entropy
Proved bound or relation: Each slice has entropy at most 4; the supremum over the whole selected m = 3 eternal flow is exactly 4 by Choi–Seo–Su–Zhao (2026), Corollary 1.5. Equality for every slice is not asserted.
Gaussian area
Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

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References on this page

  1. [AY18] Sigurd Angenent and Qian You. Ancient solutions to curve shortening with finite total curvature. 2018. arXiv:1803.01399v1. Selected theorem statements and construction passages checked November 6, 2026; proofs are cited, not independently verified. URL: https://arxiv.org/abs/1803.01399v1.All uses in this project ↩ 1
  2. [ChoiSeoSuZhao2026] Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, and Kai-Wei Zhao. Classification of ancient finite-entropy curve shortening flows. 2026. Reviewed edition 2603.09112v1; selected statements checked. URL: https://arxiv.org/abs/2603.09112v1.All uses in this project ↩ 1
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