MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.19 Castro–Lerma Lagrangian translators from curve pairs14F0E7012E88

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. Castro–Lerma — 2010-07-12 (first arXiv submission).
14F0E7012E88
B4DA2A79EC96

Castro–Lerma construct Lagrangian translating immersions in $\mathbb C^2$ from pairs of planar soliton curves. In the shrinker–expander case, choose nonvanishing arclength curves $\alpha (s),\omega (u)$ with curvature vectors $\mathbf k_\alpha =-\alpha ^\perp $ and $\mathbf k_\omega =\omega ^\perp $. Their product construction has the integral form

\[ F(s,u)=\left(\int ^ u\omega '(v)\overline{\omega (v)}\, dv- \int ^ s\alpha '(v)\overline{\alpha (v)}\, dv,\; \alpha (s)\omega (u)\right). \]

The translator velocity in this normalization is $(1,0)\in \mathbb C^2$. The induced metric is $(|\alpha |^2+|\omega |^2)(ds^2+du^2)$, so complete profile parameters bounded away from simultaneous zero give complete immersions. The more general rotating/dilating curve pairs are treated in Proposition 2; they are not all asserted to be embedded or proper. The flow is $F+t(1,0)$ for $t\in \mathbb R$. Choosing a circular shrinker and a nonstraight expander provides a concrete numerical example. The already listed Hamiltonian-stationary translator and Joyce–Lee–Tsui overlap are not new examples here. The visualized two-dimensional sheet is a projection from four real dimensions, with coordinate choice shown in the controls.

Sources: [CastroLerma2012].

Castro--Lerma Lagrangian translators from curve pairs. Numerical planar shrinker–expander pair inserted into the exact integral construction. The displayed sheet is a projection from ℂ²; projection crossings do not determine embeddedness.
Numerical ODE and integral parametrization. Numerical planar shrinker–expander pair inserted into the exact integral construction. The displayed sheet is a projection from ℂ²; projection crossings do not determine embeddedness. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Complete nonvanishing profile pairs in ℂ² · Euclidean space ℝ⁴ · Depends on the construction
Ambient space
Euclidean space ℝ⁴
Topology
Profile-product parameter surface, with any stated period identifications
Embeddedness
Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness
Noncompact
Surface genus
Not recorded.
Topological ends
Not recorded for the full parameter manifold.
Entropy
Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area
Not recorded: No quantitative conclusion is recorded from the cited construction.

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References on this page

  1. [CastroLerma2012] Ildefonso Castro and Ana M. Lerma. Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane. International Journal of Mathematics, 23:1250101, 2012. URL: https://arxiv.org/abs/1007.1886, arXiv:1007.1886, doi:10.1142/s0129167x12501017.All uses in this project ↩ 1
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