MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.18 Numerical four-handled cone shrinker candidate (Chopp, 1994; Angenent–Ilmanen–Chopp, 1995)5684D97D637F

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. Chopp — 1994 (publication year).
  2. Angenent–Ilmanen–Chopp — 1995 (publication year).
5684D97D637F
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(The numerical Angenent–Ilmanen–Chopp surface) This is Chopp's numerical genus-three, two-ended shrinker candidate in $\mathbb R^3$. It is a surface of dimension two and codimension one, with intended topology a genus-three orientable surface minus two points. The four connecting necks do not give genus four. With its axis taken to be $x_1$, the symmetry group is $\mathbb Z_2\times D_4$, where $D_4$ has eight elements and acts on the transverse plane.

The starting geometry is a cylinder with two transverse holes bored through its axis. Level selection balances radial and longitudinal neck pinches. Dilation then exposes a candidate whose two ends appear smoothly asymptotic to a nonrotational double cone $C$. The intended surface is smooth, embedded, complete, proper and without boundary, but its existence and asymptotics are numerical assertions in the cited papers. There is no closed formula. The normalization is

\[ \vec H=-\frac{X^\perp }{2},\qquad M_ t=\sqrt{-t}\, \Sigma \quad (t<0),\qquad M_0=C. \]

Here the last equality describes the proposed singular trace, not a smooth slice. Chopp's displayed mesh requires a further dilation to obtain $\Sigma =M_{-1}$.

The 1994 paper reports $E=0.0505$ and minimum cone opening about $1.2627$ radians. The 1995 paper reports approximately $1.263$ radians ($72.3$ degrees). These are numerical estimates without certified error bounds. The reported opening exceeds the numerically estimated critical rotational-expander aperture $66.04$ degrees. The barrier argument in the 1995 paper then predicts two distinct smooth self-expanding continuations for $t>0$ and fattening of the level-set flow. Its Main Result is explicitly conditional on Assertion 1 and verification of the numerics; it is not an unconditional existence proof for this particular genus-three surface. The level-set flow itself is unique as a set-valued flow; “nonuniqueness” here concerns distinct thin continuations inside it.

Sources: [Chopp1994], Section 4.6, pp. 12–13, Figures 26–28; [AIC95], Section 1 and Assertion 1, pp. 1940–1944, Section 3, pp. 1953–1955, and the conditional Main Result in Section 4, p. 1956. No uniqueness, stability spectrum, Gaussian area or entropy value is established for the numerical shrinker. The double-cone expander results are recorded in Examples 4.5 and 4.6. The rigorous Ilmanen–White family is separate (Example 2.34); its theorem does not identify a member with this computed mesh.

The rotatable picture is a genus-three topology schematic with four necks and two ends. It neither solves the shrinker equation nor verifies fattening. Chopp's related compact four-handled dumbbell in Section 4.5, Figure 24, failed his self-similarity test ($E=0.584$) and is not a sixth shrinker candidate. Semiregular polyhedra and a trefoil are future-work proposals in Section 5, not constructions established there.

Original topology schematic: genus 3, 2 ends. Chopp’s paper reports a numerical candidate; this drawing is not his mesh and does not solve the shrinker equation. Scale and neck shapes are illustrative. Outer rims and cutaway edges are display boundaries.
Schematic of a numerical candidate. Original topology schematic: genus 3, 2 ends. Chopp’s paper reports a numerical candidate; this drawing is not his mesh and does not solve the shrinker equation. Scale and neck shapes are illustrative. Outer rims and cutaway edges are display boundaries. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Numerical shrinker candidate, not a proved existence result for this computed surface. Topology and compactness describe the intended surface; the drawing is schematic.

Numerical candidate: Chopp four-handled cone candidate · Euclidean space ℝ³ · Not recorded
Ambient space
Euclidean space ℝ³
Topology
Intended orientable genus-3 surface minus 2 points
Embeddedness
Not recorded: The target is embedded and the numerical construction uses embedded level sets, but existence of an exact embedded shrinker with these properties is not proved in the reviewed computation.
Compactness
Noncompact
Surface genus
3, the topology of the numerical candidate
Topological ends
2, for the intended complete surface; finite-window boundaries are not ends
Entropy
Not recorded: No entropy value or bound established for this numerical candidate. The self-similarity diagnostic E is not entropy.
Gaussian area
Not recorded: No normalized centered Gaussian area recorded; do not infer it from the drawing or identify it with entropy for an approximate surface.

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References on this page

  1. [Chopp1994] David L. Chopp. Computation of self-similar solutions for mean curvature flow. Experimental Mathematics, 3(1):1–15, 1994. Journal scan reviewed in full. Five numerical candidates in Sections 4.1–4.4 and 4.6; the four-handled dumbbell in Section 4.5 fails the self-similarity test. No existence proof is supplied. URL: https://doi.org/10.1080/10586458.1994.10504572, doi:10.1080/10586458.1994.10504572.All uses in this project ↩ 1
  2. [AIC95] Sigurd Angenent, Tom Ilmanen, and David L. Chopp. A computed example of nonuniqueness of mean curvature flow in R3. Communications in Partial Differential Equations, 20(11–12):1937–1958, 1995. Supplied PDF has a publisher cover; article p. 1937 is PDF p. 2. Section 2 proves qualitative expander results; the genus-three shrinker is numerical Assertion 1 and the Main Result in Section 4 is conditional. URL: https://doi.org/10.1080/03605309508821158, doi:10.1080/03605309508821158.All uses in this project ↩ 1
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