MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.43 Compact periodic Lagrangian quadric shrinkers6AFBAD1A608F

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).
6AFBAD1A608F These are the compact periodic solutions in the Joyce–Lee–Tsui construction.
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(Periodic quadric shrinkers) Fix $n\geq 2$, $a_ j>0$, $\alpha =-1/2$, and put $Q(u)=\prod _{j=1}^ n(a_ j+u)$ and $\phi =\sum _ j\phi _ j$. Solve

\[ u'=2\sqrt{Q(u)}\cos (\phi -\theta ),\qquad \phi _ j'=-\frac{\sqrt{Q(u)}\sin (\phi -\theta )}{a_ j+u},\qquad \theta '=-\tfrac 12\sqrt{Q(u)}\sin (\phi -\theta ). \]

Require $a_ j+u>0$ and $A=\sqrt{Q(u)}e^{-u/4}\sin (\phi -\theta )>0$. Choose nonconstant periodic data for which all phase increments over a period of $u$ are rational multiples of $\pi $. A common period $T$ then closes all $w_ j(s)=\sqrt{a_ j+u(s)}e^{i\phi _ j(s)}$. The immersion is

\[ X(x,s)=(x_1w_1(s),\ldots ,x_ nw_ n(s)),\qquad (x,s)\in \mathbb S^{n-1}\times (\mathbb R/T\mathbb Z). \]

Theorems A, B, E(a)(ii), and F of [JoyceLeeTsui2010] establish these ODE solutions and a dense set of closing initial data. They satisfy $\vec H=-X^\perp /2$, and $\sqrt{-t}\, X$, $t<0$, is their mean curvature flow.

The parameter manifold is compact, hence the immersion is complete and proper. It has dimension and codimension $n$, no ends, and topology $\mathbb S^{n-1}\times \mathbb S^1$; for $n=2$ it is a torus. Coordinate reflections are symmetries. Embeddedness, uniqueness, and stability are not inferred from the immersion theorem. The drawing is a construction diagram, not a numerically closed solution. The constant-radius subcase, including familiar overlapping examples, is excluded here.

The ODE and rational phase-closing conditions produce a compact immersion with domain Sⁿ⁻¹ × S¹. No numerical closed orbit or embeddedness is claimed by this picture.
Construction diagram. The ODE and rational phase-closing conditions produce a compact immersion with domain Sⁿ⁻¹ × S¹. No numerical closed orbit or embeddedness is claimed by this picture. 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.

Nonconstant periodic quadric shrinker in C² · Euclidean space ℝ⁴ · Not recorded
Ambient space
Euclidean space ℝ⁴
Topology
Torus S¹ × S¹ (immersion domain)
Embeddedness
Not recorded: The theorem establishes an immersion of Sⁿ⁻¹ × S¹. It does not establish embeddedness for every closing datum; the constant-radius subcase is excluded.
Compactness
Compact
Surface genus
1 for the torus parameter domain in C².
Topological ends
No ends (compact domain).
Entropy
Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.
Gaussian area
Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.
Nonconstant periodic quadric shrinker in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Not recorded
Ambient space
Euclidean dimension ≥ 5
Topology
Sⁿ⁻¹ × S¹
Embeddedness
Not recorded: The theorem establishes an immersion of Sⁿ⁻¹ × S¹. It does not establish embeddedness for every closing datum; the constant-radius subcase is excluded.
Compactness
Compact
Surface genus
Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends
No ends (compact domain).
Entropy
Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.
Gaussian area
Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.

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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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