$\mathbb{R}$-equivariant minimal Lagrangian surfaces in the complex quadric $Q_2$ and the complex hyperbolic quadric $Q_2^{\ast }$

Abstract

We study global and variational properties of $\mathbb{R}$-equivariant minimal Lagrangian surfaces in the complex quadric $Q_2$ and the complex hyperbolic quadric $Q_2^{\ast }$ arising from loop group potentials. In the $Q_2^{\ast }$ case, we work under the timelike-axis condition throughout the associated ${\mathbb{S}}^1$-family. Excluding the flat case in $Q_2$, we give a necessary and sufficient criterion for closing in the $y$-direction: a member $f^{{\lambda }_0}$ closes if and only if the spectral ratio ${\mu }_2/{\mu }_1$ is a positive rational number. The closing parameters form a dense countable subset of ${\mathbb{S}}^1$, while the non-closing parameters are dense and have full arc-length measure. The closing members yield cylindrical quotients in $Q_2$ and annular quotients in $Q_2^{\ast }$. Apart from the flat case in $Q_2$, the surfaces under consideration have infinite absolute total curvature. The equivariant surfaces considered here in $Q_2$ are unstable, whereas minimal Lagrangian surfaces in $Q_2^{\ast }$ are stable. Finally, for compact cylindrical domains of the $y$-closing surfaces in $Q_2$, we obtain explicit sufficient width conditions for Hamiltonian stability and instability under variations fixed near the boundary. We also establish the existence of a finite geometric critical width and determine it explicitly in the flat case.