MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.18 Explicit rotating Lagrangian translators5C21E6709CA8

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).
5C21E6709CA8 An elementary subfamily of the same quadric ansatz has unbounded Lagrangian angle.
A19CAED49B0D

(Rotating translating planes) For $n\geq 2$, put $k=n-1$. The exact map

\[ X(x,s)=\left(x_1e^{is},\ldots ,x_ ke^{is},\tfrac 12|x|^2-is\right), \qquad (x,s)\in \mathbb R^ k\times \mathbb R, \]

is a complete proper embedded Lagrangian in $\mathbb C^ n$, of dimension and codimension $n$. Its translator velocity is $V=(0,\ldots ,0,k)$, so $X+tV$, $t\in \mathbb R$, is eternal. Spatial dilation by $k$ gives unit translating speed if desired.

This is the constant-radius case of Theorem G and Corollary H of [JoyceLeeTsui2010], with $\lambda _ j=-1$, $a_ j=A=1$, $u=0$, and $\alpha =k$. Directly, the metric and a lifted angle are

\[ g=\sum _ jdx_ j^2+\left(\sum _ jx_ jdx_ j\right)^2+(1+|x|^2)ds^2, \qquad \theta =ks-\pi /2. \]

Consequently

\[ \vec H=J\nabla \theta =\frac{k}{1+|x|^2}(-x_1e^{is},\ldots ,-x_ ke^{is},1)=V^\perp . \]

The last imaginary coordinate recovers $s$, proving injectivity; $|X|^2=|x|^2+|x|^4/4+s^2$ proves properness. The metric dominates the Euclidean domain metric, proving completeness. Topologically it is $\mathbb R^ n$, with one end. It has a combined rotation and imaginary-translation symmetry and is not almost calibrated. No stability or entropy value is claimed. The four-dimensional illustration samples the exact $n=2$ formula.

The n = 2 map (x eⁱˢ, x²/2 − is) on a finite parameter window. Its last imaginary coordinate separates turns. Use 4D rotation to change the projection.
Exact four-dimensional projection. The n = 2 map (x eⁱˢ, x²/2 − is) on a finite parameter window. Its last imaginary coordinate separates turns. Use 4D rotation to change the projection. 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
Finite; value not recorded: The exact parametrization has area density 1+x² and |X|²=x²+x⁴/4+s², so the centered Gaussian integral is finite. This does not assert finite entropy.
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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