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 ${\nabla^λ}_{λ\in \mathbb S^{1}}$ on a trivial bundle. We prove that minimality is equivalent to the flatness of $\nabla^λ$ for all $λ$, 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.

💡 Research Summary

This paper develops a DPW‑type loop‑group representation for minimal Lagrangian surfaces in the complex quadric (Q_{2}), which is isometric to the product (S^{2}\times S^{2}). Starting from a Lagrangian conformal immersion (f\colon M\to Q_{2}) of a Riemann surface, the authors choose a horizontal lift (\tilde f\colon D\subset M\to S^{7}\subset\mathbb C^{4}) and construct an (SO(4)) moving frame (F). The Maurer–Cartan form (\omega=F^{-1}dF) takes values in (\mathfrak{so}(4)=\mathfrak{k}\oplus\mathfrak{p}), where (\mathfrak{k}\cong\mathfrak{so}(2)\times\mathfrak{so}(2)) is the isotropy algebra of the symmetric space (Q_{2}=SO(4)/SO(2)\times SO(2)).

A one‑parameter family of connections is introduced: \