MCF-stack: Special Solutions of Mean Curvature Flow

§ 4.13 Lagrangian expander necks between transverse planes (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)6095D0C72380

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).
6095D0C72380 These expanders smooth two transverse Lagrangian planes in $\mathbb C^ n$.
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(Expanding Lagrangian necks) Fix $n\geq 2$, $a_ j>0$, and $\psi _ j\in \mathbb R$. Set

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

With $\alpha =1/2$, the exact integral parametrization

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

satisfies $\vec H=X^\perp /2$ and gives the flow $\sqrt t\, X$, $t>0$. By [JoyceLeeTsui2010], Theorems C–D, its image is a proper embedded Lagrangian neck. It is complete, has dimension and codimension $n$, and has two ends. For $n=2$ its topology is an annulus; generally it is $\mathbb S^{n-1}\times \mathbb R$.

Write $\bar\phi _ j=\phi _ j(+\infty )-\psi _ j$. The limiting planes have coordinate phases $\psi _ j\pm \bar\phi _ j$, with $0<2\sum _ j\bar\phi _ j<\pi $. Every admissible angle tuple determines unique positive $a_ j$ at fixed $\alpha $ within this family. The neck is zero-Maslov and almost calibrated. Equal $a_ j$ and equal $\psi _ j$ give rotational symmetry under $\mathrm{SO}(n)$.

For $n\geq 3$, the stronger uniqueness theorem in [ImagiJoyceOliveira2016], Theorem 1.2, requires an embedded, exact, closed-as-a-subset Lagrangian expander asymptotic to two transverse planes at rate $\rho <2$. These hypotheses must not be dropped. No general stability claim is attached. The figure projects the $n=2$, $a_1=a_2=1$, $\psi _1=\psi _2=0$ member, evaluating its integrals numerically.

The exact integral parametrization in C² with a₁ = a₂ = 1 and α = 1/2, evaluated numerically. Both planar ends are truncated. This is a projection, not an R³ expander.
Numerical quadrature and projection. The exact integral parametrization in C² with a₁ = a₂ = 1 and α = 1/2, evaluated numerically. Both planar ends are truncated. This is a projection, not an R³ expander. 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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