MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.1 Closed planar shrinking curves (Abresch–Langer, 1986; Halldórsson, 2010; Drugan–Lee–X. H. Nguyen, 2017; Colding–Minicozzi II, 2019)B4D1BD157D4B

Earliest recorded reference: (publication year).

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. Abresch–Langer — 1986 (publication year).
  2. Halldórsson — 2010-07-09 (first arXiv submission).
  3. Drugan–Lee–X. H. Nguyen — 2017-08-30 (first arXiv submission).
  4. Colding–Minicozzi II — 2019-03-08 (first arXiv submission).
B4D1BD157D4B

The closed planar curves are classified by [AL1986]. The supplied sources record the curvature conservation laws in [DLN2018], Theorem 1, PDF pp. 5–6, and the rotation-index description in [CMComplexity2020], Section 6.3, supplied PDF p. 22.

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Besides the round circle, there are noncircular closed immersed curves $\gamma _{m,\ell }\subset \mathbb R^2$, indexed primitively by coprime positive integers satisfying

\[ \frac12<\frac m\ell <\frac1{\sqrt2}. \]

Here $m$ is the rotation index and $\ell $ is the number of periods of the curvature. Their flows are $\sqrt{-t}\, \gamma _{m,\ell }$ for $t<0$. An ODE description in the normalization of this collection is

\[ \kappa _{\theta \theta }+\kappa =\frac1{2\kappa },\qquad \kappa >0,\qquad \kappa _\theta ^2+\kappa ^2-\log \kappa =E. \]

Choose the nonconstant periodic solution with curvature period $2\pi m/\ell $. With $T(\theta )=(\cos \theta ,\sin \theta )$ and $N(\theta )=(-\sin \theta ,\cos \theta )$, reconstruction gives

\[ X(\theta )=2\kappa _\theta T(\theta )-2\kappa N(\theta ), \qquad \theta \in \mathbb R/(2\pi m\mathbb Z). \]

This is an ODE construction with a closing condition, not a closed elementary formula for $\kappa $.

  • The domain is $S^1$. These are compact, complete, proper immersions with self-intersections. Their curvature is positive after orienting the curve, so they are locally strictly convex; their images do not bound convex embedded domains.

  • The round circle is the only embedded closed shrinking curve. The noncircular curves have finite entropy strictly greater than two, as recorded in [CMComplexity2020], Section 6.3.

  • The conservation law $\kappa e^{-|X|^2/4}=C>0$ relates curvature to distance from the shrinking center. Compactness makes the shrinking flow a Type I flow becoming extinct at the origin.

The interactive figure starts with $(m,\ell )=(2,3)$. Enter other coprime indices in the stated range, or use the example buttons, to compute more numerical profiles. The viewer supports $m,\ell \leq 100$ and reports the numerical closing gap. This display limit is not a restriction of the classification theorem. The static drawing and printed figure show $(2,3)$. These computations illustrate the closing condition; their residuals are not certified error bounds.

The nonclosed irrational-turning branch is recorded separately in Example 2.11; see also [Halldorsson2012], Theorem 5.1.

Numerical Abresch–Langer closed planar curve, initially with turning number 2 and three curvature periods. Controls allow other admissible coprime indices.
Numerical ODE profile. Choose m (turning number) and ℓ (curvature periods) to compute a numerical Abresch–Langer curve. The support function solves h″+h=2/h with period 2πm/ℓ. The fixed drawing and print version show (2, 3). Numerical closing gaps are diagnostics, not certified error bounds. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
Immersed circle S¹ · Euclidean plane ℝ² · Not embedded
Ambient space
Euclidean plane ℝ²
Topology
Immersed circle S¹
Embeddedness
Not embedded: The noncircular primitive Abresch–Langer curves recorded here have self-intersections. The embedded shrinking circle is listed separately.
Compactness
Compact
Surface genus
Not a surface
Topological ends
0 (compact)
Entropy
Proved bound or relation: λ > 2 for the noncircular closed curves.
Gaussian area
Proved bound or relation: F₀,₁ = λ for the normalized shrinker. λ > 2 for the noncircular closed curves.

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References on this page

  1. [AL1986] Uwe Abresch and Joel Langer. The normalized curve shortening flow and homothetic solutions. Journal of Differential Geometry, 23(2):175–196, 1986. Original classification; the supplied survey and Complexity of parabolic systems were also consulted. URL: https://doi.org/10.4310/jdg/1214440025, doi:10.4310/jdg/1214440025.All uses in this project ↩ 1
  2. [DLN2018] Gregory Drugan, Hojoo Lee, and Xuan Hien Nguyen. A survey of closed self-shrinkers with symmetry. Results in Mathematics, 73(1):32, 2018. Article 32, 32 pages; locators refer to the supplied published edition. URL: https://arxiv.org/abs/1708.09113, doi:10.1007/s00025-018-0763-3.All uses in this project ↩ 1
  3. [CMComplexity2020] Tobias Holck Colding and William P. Minicozzi, II. Complexity of parabolic systems. Publications Mathématiques de l'IHÉS, 132:83–135, 2020. The supplied 53-page IHES PDF is used for source page locators. URL: https://www.numdam.org/articles/10.1007/s10240-020-00117-x/, doi:10.1007/s10240-020-00117-x.All uses in this project ↩ 1 ↩ 2
  4. [Halldorsson2012] Höskuldur P. Halldórsson. Self-similar solutions to the curve shortening flow. Transactions of the American Mathematical Society, 364(10):5285–5309, 2012. URL: https://arxiv.org/abs/1007.1617v2, doi:10.1090/S0002-9947-2012-05632-7.All uses in this project ↩ 1
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