Lagrangian surfaces in $\mathbb H^2 \times \mathbb H^2$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866912556677857280 |
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| author | Gao, Dong Van der Veken, Joeri Wijffels, Anne Xu, Botong |
| author_facet | Gao, Dong Van der Veken, Joeri Wijffels, Anne Xu, Botong |
| contents | The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural Kähler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study Lagrangian surfaces in this manifold. We present several examples and classify the totally umbilical and totally geodesic Lagrangian surfaces, the Lagrangian surfaces with parallel second fundamental form, the minimal Lagrangian surfaces with constant Gaussian curvature and the complete minimal Lagrangian surfaces satisfying a bounding condition on an important function that can be defined on any Lagrangian surface in this particular ambient space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_13975 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lagrangian surfaces in $\mathbb H^2 \times \mathbb H^2$ Gao, Dong Van der Veken, Joeri Wijffels, Anne Xu, Botong Differential Geometry 53C42, 53D12, 53B25 The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural Kähler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study Lagrangian surfaces in this manifold. We present several examples and classify the totally umbilical and totally geodesic Lagrangian surfaces, the Lagrangian surfaces with parallel second fundamental form, the minimal Lagrangian surfaces with constant Gaussian curvature and the complete minimal Lagrangian surfaces satisfying a bounding condition on an important function that can be defined on any Lagrangian surface in this particular ambient space. |
| title | Lagrangian surfaces in $\mathbb H^2 \times \mathbb H^2$ |
| topic | Differential Geometry 53C42, 53D12, 53B25 |
| url | https://arxiv.org/abs/2106.13975 |