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

2025/10/17 by Kobayashi, Shimpei, Zeng, Sihao
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.15326

Abstract

We develop a loop group (DPW-type) representation for minimal Lagrangian surfaces in the complex quadric Q₂≅ \mathbb S²× \mathbb S², formulated via a flat family of connections \∇^λ\_{λ∈ \mathbb S¹} on a trivial bundle. We prove that minimality is equivalent to the flatness of ∇^λ for all λ, describe the associated isometric \mathbb S¹-family, and establish a precise correspondence with minimal surfaces in \mathbb S³ 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.

Citations

• Equivariant primitive harmonic maps into k-symmetric spaces, with applications to Willmore surfaces
• Minimal Lagrangian surfaces in ℂP² via the loop group method Part II: The general case
• Ruh-Vilms Theorems For Minimal Surfaces Without Complex Points and Minimal Lagrangian Surfaces in \mathbb C P²
• Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres
• Hamiltonian stationary Lagrangian surfaces in C²

Related

• A new look at equivariant minimal Lagrangian surfaces in ℂP²
• Minimal Lagrangian surfaces in ℂP² via the loop group method Part II: The general case
• A loop group method for minimal surfaces in the three-dimensional Heisenberg group
• Formal Killing fields for minimal Lagrangian surfaces in complex space forms
• Timelike Minimal Surfaces via Loop Groups