MCF-stack: Special Solutions of Mean Curvature Flow

§ 4.5 Two-sheeted expanders of rotational double conesADE41440A542

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. Angenent–Ilmanen–Chopp — 1995 (publication year).
ADE41440A542
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(Disconnected double-cone smoothings) Let $n\geq 2$, write $(x,z)\in \mathbb R^ n\times \mathbb R$, and let

\[ D_\alpha =\{ (x,z):|x|=|z|\tan \alpha \} ,\qquad 0<\alpha <\pi /2. \]

With $\phi _{\cot \alpha }$ from Example 4.7, the two-sheeted expander is

\[ M_\alpha ^{(2)} =\{ (x,\phi _{\cot \alpha }(|x|)):x\in \mathbb R^ n\} \mathbin {\cup } \{ (x,-\phi _{\cot \alpha }(|x|)):x\in \mathbb R^ n\} . \]

This display specifies the surface using ODE-defined functions. Its two components are smooth, complete, properly embedded convex entire graphs, each diffeomorphic to $\mathbb R^ n$. The union is noncompact, invariant under rotations about the $z$-axis and under $z\mapsto -z$, and has two oppositely directed conical ends. The flow $\sqrt t\, M_\alpha ^{(2)}$, $t>0$, approaches $D_\alpha $ as $t\downarrow 0$.

Existence and uniqueness of each rotational single-lobe graphical profile are established in [AIC95], Lemma 2(i), printed p. 1947, supplied PDF p. 12. Theorem 4, printed pp. 1950–1951, PDF pp. 15–16, provides a critical angle $\alpha _{\rm crit}(n+1)\in (0,\pi /2)$. For $\alpha <\alpha _{\rm crit}(n+1)$ the level-set flow of the double cone is exactly the two-sheeted flow. For angles above this threshold there are also connected smoothings, so the graphical uniqueness statement does not assert uniqueness among all flows from the cone. The critical-angle decimal estimates later in that paper are numerical. The $n=2$ figure rotates two reflected numerical profiles; the cone is their asymptotic limit.

Two-sheeted expanders of rotational double cones. Two reflected rotational graphical sheets with tip heights +1 and -1. The sheets share a double cone at infinity and are truncated at radius 6.
Numerical ODE surface. Two reflected rotational graphical sheets with tip heights +1 and -1. The sheets share a double cone at infinity and are truncated at radius 6. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Disconnected: two components. Genus is recorded componentwise.

Two disjoint planes ℝ² ⊔ ℝ² · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Two disjoint planes ℝ² ⊔ ℝ²
Embeddedness
Embedded: The two smooth graphical components are disjoint; their disconnected union is embedded.
Compactness
Noncompact
Surface genus
0 on each component
Topological ends
2 total; 1 on each component
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.
Higher-dimensional double-cone smoothings · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Two disjoint copies of ℝⁿ, n ≥ 3
Embeddedness
Embedded: The two smooth graphical components are disjoint; their disconnected union is embedded.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
2 total; 1 on each component
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.

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References on this page

  1. [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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