MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.17 Joyce–Lee–Tsui small-angle Lagrangian translators16AF28E8CCDB

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. Joyce–Lee–Tsui — 2008-01-24 (first arXiv submission).
16AF28E8CCDB The following unit-speed translators have dimension and codimension $n$ in $\mathbb C^ n$.
BCD2A9973070

(Almost-calibrated translating Lagrangians) Fix $n\geq 2$ and $a_1,\ldots ,a_{n-1}>0$. Define the positive smooth function

\[ P(y)=\frac{e^{y^2}\prod _{j=1}^{n-1}(1+a_ jy^2)-1}{y^2}, \qquad P(0)=1+\sum _{j=1}^{n-1}a_ j, \]

and the quadratures

\[ \phi _ j(y)=\int _0^ y\frac{dt}{(a_ j^{-1}+t^2)\sqrt{P(t)}},\qquad B(y)=\int _0^ y\frac{dt}{\sqrt{P(t)}}. \]

For $(x,y)\in \mathbb R^{n-1}\times \mathbb R$, set

\[ X(x,y)=\left(x_ j\sqrt{a_ j^{-1}+y^2}\, e^{i\phi _ j(y)}\right)_{j=1}^{n-1} \mathbin {\times }\left(\tfrac 12(y^2-|x|^2)+iB(y)\right). \]

Here the displayed product concatenates complex coordinates. This is Corollary I of [JoyceLeeTsui2010] with $\alpha =1$, after a constant imaginary translation in the last coordinate. Differentiating the source's angle expression gives $\theta '=-1/\sqrt P$, so this integral form avoids a choice of argument branch.

The image is a complete proper embedded Lagrangian $\mathbb R^ n$ with one topological end. It satisfies $\vec H=V^\perp $ for $V=(0,\ldots ,0,1)$ and evolves as $X+tV$ for all $t\in \mathbb R$. If $\bar\phi _ j=\lim _{y\to \infty }\phi _ j(y)$, then $0<\sum _ j\bar\phi _ j<\pi /2$ and the angle oscillation is $\pi -2\sum _ j\bar\phi _ j$; it can be arbitrarily small. The plane-pair description in the source is heuristic, not an asserted smooth asymptotic-cone theorem. The figure uses numerical quadrature of this exact integral parametrization in $\mathbb C^2$, followed by projection.

The n = 2, a₁ = α = 1 exact integral parametrization is evaluated numerically and projected from R⁴. The displayed patch is truncated; the full embedded surface has one end.
Numerical quadrature and projection. The n = 2, a₁ = α = 1 exact integral parametrization is evaluated numerically and projected from R⁴. The displayed patch is truncated; the full embedded surface has one end. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian translator in C² · Euclidean space ℝ⁴ · Embedded
Ambient space
Euclidean space ℝ⁴
Topology
R²
Embeddedness
Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness
Noncompact
Surface genus
0 (plane topology).
Topological ends
One end of Rⁿ, n ≥ 2.
Entropy
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Lagrangian translator in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Rⁿ
Embeddedness
Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness
Noncompact
Surface genus
Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends
One end of Rⁿ, n ≥ 2.
Entropy
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.

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References on this page

  1. [JoyceLeeTsui2010] Dominic Joyce, Yng-Ing Lee, and Mao-Pei Tsui. Self-similar solutions and translating solitons for lagrangian mean curvature flow. 2010. Reviewed arXiv edition 0801.3721v2. Theorems A–F, Corollaries H–I, Sections 3–6. Periodic compact case E(a)(ii); plane-neck cases C–D; translator formula (18), with direct integral rewrite; constant-radius rotating subcase of G/H. URL: https://arxiv.org/abs/0801.3721v2.All uses in this project ↩ 1
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