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arXiv · 2605.17876

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

Shimpei Kobayashi·Sihao Zeng
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Abstract

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.

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Shimpei Kobayashi, Sihao Zeng. 2026-09-03. Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method. https://arxiv.org/abs/2605.17876

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