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Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method
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 π1-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 β-equivariant and radially symmetric surfaces. In particular, under suitable conditions, the β-equivariant family contains catenoid-type examples.
Related
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BibTeX
@misc{kobayashi2026minimal,
title = {{Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method}},
author = {Shimpei Kobayashi and Sihao Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.17876},
eprint = {2605.17876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17876},
}
RIS
TY - UNPB
TI - Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method
AU - Shimpei Kobayashi
AU - Sihao Zeng
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2605.17876
UR - https://arxiv.org/abs/2605.17876
AB - 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 π^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 β-equivariant and radially symmetric surfaces. In particular, under suitable conditions, the β-equivariant family contains catenoid-type examples.
ER -