§ 2.49 Lagrangian shrinkers with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)689C0F332420

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 $\lambda =1/2$, there are closed profiles indexed by coprime positive $p,q$ with $1/(2n)<p/q<1/\sqrt{2n}$. The domain $S^1\times M$ is compact and the immersion is complete and proper. Its image can involve finite identifications; the product domain must not be confused with the topology of an embedded image. 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].