Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method

We study minimal Lagrangian surfaces in the complex hyperbolic quadric. We show that minimality of a Lagrangian surface is characterized by a loop of flat connections, which yields an associated ${\mathbb{S}}^1$-family of isometric deformations. We also establish a correspondence with spacelike maximal surfaces in anti-de Sitter $3$-space via the Gauss map. Using the resulting harmonic map into the hyperbolic two-space, we develop a DPW-type representation and construct explicit examples, including $\mathbb{R}$-equivariant and radially symmetric surfaces. In particular, under suitable conditions, the $\mathbb{R}$-equivariant family contains catenoid-type examples.