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

We develop a loop group (DPW-type) representation for minimal Lagrangian surfaces in the complex quadric $Q_2\cong {\mathbb{S}}^2\times {\mathbb{S}}^2$, formulated via a flat family of connections $\{{\mathrm{\nabla }}^{\lambda }{\}}_{\lambda \in {\mathbb{S}}^1}$ on a trivial bundle. We prove that minimality is equivalent to the flatness of ${\mathrm{\nabla }}^{\lambda }$ for all $\lambda$, describe the associated isometric ${\mathbb{S}}^1$-family, and establish a precise correspondence with minimal surfaces in ${\mathbb{S}}^3$ through their Gauss maps. Our framework unifies and streamlines earlier constructions (e.g., Castro–Urbano) and yields explicit families including $\mathbb{R}$-equivariant, radially symmetric, and trinoid-type examples.