MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.5 The Angenent oval as an explicit ancient flow34E98635F82B

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. Broadbridge–Vassiliou — 2011-06-01 (first arXiv submission).
34E98635F82B
B008243C5D96

(The ancient paperclip curve) With the speed-one curve-shortening convention, the exact implicit equation

\[ \Gamma _ t=\{ (x,y)\in \mathbb R^2:|x|<\pi /2,\ e^ t\cosh y=\cos x\} , \qquad -\infty <t<0, \]

defines the Angenent oval. Equivalently, its two graph branches are

\[ y=\pm \operatorname {arcosh}(e^{-t}\cos x), \qquad |x|\leq \arccos (e^ t). \]

Their endpoints join smoothly as a curve, even though those graph charts cease to be regular there. This is an exact implicit formula, rather than a single global graph parametrization. It is recorded in [BV11], Section 3.1, equation (14), p. 7.

Every time slice is a smooth compact embedded strictly convex curve, hence complete and proper, with topology $S^1$ and reflection symmetries across both coordinate axes. It shrinks to the origin at time zero but changes shape and is not a soliton. Its enclosed area is $-2\pi t$, and

\[ \kappa _{\max }=(1-e^{2t})^{-1/2},\qquad \kappa _{\min }=e^ t(1-e^{2t})^{-1/2}. \]

The rescaling $(-t)^{-1/2}\Gamma _ t$ converges smoothly to the radius-$\sqrt2$ circle at extinction. As $t\to -\infty $, translating its upper and lower branches to keep their tips fixed gives the two oppositely directed grim-reaper profiles, locally in $|x|<\pi /2$. These limits can be read from the implicit equation: near extinction its quadratic expansion gives $x^2+y^2\sim -2t$, and subtracting the upper tip height gives the limiting graph $\log \cos x$ as $t\to -\infty $. The time slider plots the exact implicit curve for $-5\leq t\leq -0.1$ on a fixed spatial scale.

The Angenent oval as an explicit ancient flow. The planar ancient oval satisfies cos(x)=exp(t) cosh(y). The time slider shows finite negative times; its smooth flow exists for all t<0.
Exact implicit solution. The planar ancient oval satisfies cos(x)=exp(t) cosh(y). The time slider shows finite negative times; its smooth flow exists for all t<0. The time slider uses a fixed spatial scale. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

The entry does not currently give a slice-by-slice entropy value.

Circle S¹ · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Circle S¹
Embeddedness
Embedded: Every time slice for t < 0 is an embedded strictly convex closed curve.
Compactness
Compact
Surface genus
Not a surface
Topological ends
0 (compact)
Entropy
Finite; value not recorded: Finite; no value recorded.
Gaussian area
Not recorded: Not recorded in this entry.

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References on this page

  1. [BV11] Philip Broadbridge and Peter Vassiliou. The role of symmetry and separation in surface evolution and curve shortening. SIGMA, 7:052, 2011. 19 pages; published June 1, 2011. URL: https://www.emis.de/journals/SIGMA/2011/052/sigma11-052.pdf, doi:10.3842/SIGMA.2011.052.All uses in this project ↩ 1
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