§ 4.17 Lagrangian expanders with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)7681DAB7225B

Let $\psi :M^{n-1}\to S^{2n-1}$ be a compact minimal Legendrian immersion. For an arclength curve $\gamma :I\to \mathbb C\setminus \{ 0\} $, the map

is Lagrangian in $\mathbb C^ n$. It satisfies $\mathbf H+\lambda F^\perp =0$ precisely when

Writing $\gamma =re^{i\phi }$ and $\alpha $ for tangent angle minus $\phi $, the profile equations are $r'=\cos \alpha $, $\phi '=\sin \alpha /r$, $\alpha '=(\lambda r-n/r)\sin \alpha $.

For $\lambda =-1/2$, Anciaux gives a one-parameter family of complete embedded profile curves with two asymptotic rays at angle in $(0,\pi /n)$. The full immersion has domain $\mathbb R\times M$ and two ends for connected compact $M$. It is complete and proper; full embeddedness depends on $\psi $ and possible identifications between leaves, not just on embeddedness of the planar profile. Its flow is $\sqrt t\, F$, $t>0$.

This entry emphasizes non-spherical minimal Legendrian leaves, possible for $n\ge 3$, including the admissible group orbits in Madnick–Wood. The spherical-leaf overlap with the existing quadric or neck examples is not counted again. The figure uses the minimal Legendrian two-torus $\psi (u,v)=3^{-1/2}(e^{iu},e^{iv},e^{-i(u+v)})$ and a numerical radial profile, with a labeled projection of a two-dimensional slice of the threefold in $\mathbb C^3$. Apparent projection intersections do not prove nonembeddedness.

Sources: [Anciaux2006, MadnickWood2025].