MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.28 Stationary special Lagrangian necks (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)380EC54D3A30

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).
  2. Imagi–Joyce–Oliveira dos Santos — 2014-04-01 (first arXiv submission).
380EC54D3A30 Lawlor necks are the stationary endpoint of the Lagrangian neck construction.
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(Stationary special Lagrangian necks) For $n\geq 2$, $a_ j>0$, and phases $\psi _ j\in \mathbb R$, define

\[ P_0(y)=\frac{\prod _{j=1}^ n(1+a_ jy^2)-1}{y^2},\quad P_0(0)=\sum _ ja_ j,\quad \phi _ j(y)=\psi _ j+\int _0^ y\frac{dt}{(a_ j^{-1}+t^2)\sqrt{P_0(t)}}. \]

The map

\[ X(x,y)=\left(x_ j\sqrt{a_ j^{-1}+y^2}e^{i\phi _ j(y)}\right)_{j=1}^ n, \qquad (x,y)\in \mathbb S^{n-1}\times \mathbb R, \]

parametrizes a complete proper embedded special Lagrangian of dimension and codimension $n$. Its constant flow $L_ t=X(\mathbb S^{n-1}\times \mathbb R)$ is eternal, $t\in \mathbb R$, since $\vec H=0$. This is the $\alpha =0$ case of [JoyceLeeTsui2010], Theorems C–D, credited there to Lawlor.

It has two planar ends and topology $\mathbb S^{n-1}\times \mathbb R$; in $\mathbb C^2$ it is an annulus. The positive characteristic angles sum to $\pi $. They determine a neck up to scale, and equal $a_ j$ give the rotational subfamily after a unitary change of coordinates. Calibration makes the neck area-minimizing for compactly supported competitors with the same boundary, hence stable for compactly supported variations.

For $n\geq 3$, [ImagiJoyceOliveira2016], Theorem 1.1, classifies closed-as-subsets, embedded, exact, asymptotically conical special Lagrangians with two transverse planar asymptotes and rate $\rho <0$, up to the specified special unitary rotation. The source's weaker rate $\rho <2$ additionally permits translation. These are conditional uniqueness statements. The drawing projects the exact $n=2$, $a_1=a_2=1$ map, whose common complex factor is $\sqrt{1+y^2/2}+iy/\sqrt2$.

A truncated exact Lawlor annulus in C² with a₁ = a₂ = 1. It is stationary under MCF. Projected crossings do not change its embeddedness in R⁴.
Exact four-dimensional projection. A truncated exact Lawlor annulus in C² with a₁ = a₂ = 1. It is stationary under MCF. Projected crossings do not change its embeddedness in R⁴. 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 neck in C² · Euclidean space ℝ⁴ · Embedded
Ambient space
Euclidean space ℝ⁴
Topology
Annulus S¹ × R
Embeddedness
Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness
Noncompact
Surface genus
0 (annulus).
Topological ends
Two planar ends.
Entropy
Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.
Gaussian area
Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.
Lagrangian neck in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Sⁿ⁻¹ × R
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
Two planar ends.
Entropy
Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.
Gaussian area
Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.

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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
  2. [ImagiJoyceOliveira2016] Yohsuke Imagi, Dominic Joyce, and Joana Oliveira dos Santos. Uniqueness results for special lagrangians and lagrangian mean curvature flow expanders in cm. 2016. Reviewed arXiv edition 1404.0271v2. Theorems 1.1–1.2, Remark 1.3, Examples 3.9 and 3.15; n ≥ 3, exactness, embeddedness and asymptotic rates retained. URL: https://arxiv.org/abs/1404.0271v2.All uses in this project ↩ 1
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