Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method
Shimpei Kobayashi, Sihao Zeng
TL;DR
This work formulates a DPW-type loop group representation for minimal Lagrangian surfaces in the complex quadric $Q_{2}$ by encoding the surface data in a flat one-parameter family of connections $\{\nabla^{\lambda}\}_{\lambda\in\mathbb{S}^{1}}$. Minimality is characterized by the flatness of the whole family, linking holomorphic invariants and the elliptic sinh-Gordon structure to an associated harmonic map into $\mathbb{S}^{2}$ and to Gauss maps of minimal surfaces in $\mathbb{S}^{3}$; this yields a generalized Weierstrass representation (DPW) for $Q_{2}$ via the isomorphism $\mathrm{SO}(4)\cong(\mathrm{SU}(2)\times\mathrm{SU}(2))/\mathbb{Z}_{2}$. The approach unifies Castro–Urbano’s framework and enables explicit families, including $\mathbb{R}$-equivariant, radially symmetric, and trinoid-type minimal Lagrangian surfaces in $Q_{2}$, constructed from holomorphic potentials and analyzed through monodromy and closing conditions. This DPW construction also clarifies the correspondence with minimal surfaces in $\mathbb{S}^{3}$ via the Sasaki-like metric on the unit tangent bundle and provides a practical pipeline to generate new examples in the Lagranangian setting. Overall, the paper advances integrable-systems techniques for complex-quadric geometry and broadens the catalog of explicit minimal Lagrangian surfaces in $Q_{2}$.
Abstract
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.