Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914297408389120 |
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| author | Kobayashi, Shimpei Zeng, Sihao |
| author_facet | Kobayashi, Shimpei Zeng, Sihao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method Kobayashi, Shimpei Zeng, Sihao Differential Geometry Primary 53C42, Secondary 53D12 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. |
| title | Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method |
| topic | Differential Geometry Primary 53C42, Secondary 53D12 |
| url | https://arxiv.org/abs/2510.15326 |