A classification of constant Gaussian curvature surfaces in the three-dimensional hyperbolic space
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909884899917824 |
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| author | Inoguchi, Junichi Kobayashi, Shimpei |
| author_facet | Inoguchi, Junichi Kobayashi, Shimpei |
| contents | We classify weakly complete constant Gaussian curvature $-1<K<0$ surfaces in the hyperbolic three-space in terms of holomorphic quadratic differentials. For this purpose, we first establish a loop group method for constant Gaussian curvature surfaces with $K>-1$ and $K \neq 0$ via the harmonicity of the Lagrangian and Legendrian Gauss maps. We then show that a spectral parameter deformation of the Lagrangian harmonic Gauss map gives a harmonic map into the hyperbolic two-space for $-1< K<0$ or the two-sphere for $K>0$, respectively. Consequently, weakly complete constant Gaussian curvature surfaces with $-1 < K <0$ are in one-to-one correspondence with holomorphic quadratic differentials on the unit disk or the complex plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A classification of constant Gaussian curvature surfaces in the three-dimensional hyperbolic space Inoguchi, Junichi Kobayashi, Shimpei Differential Geometry Primary 53C42, 58D10, Secondary 53C43, 53A10 We classify weakly complete constant Gaussian curvature $-1<K<0$ surfaces in the hyperbolic three-space in terms of holomorphic quadratic differentials. For this purpose, we first establish a loop group method for constant Gaussian curvature surfaces with $K>-1$ and $K \neq 0$ via the harmonicity of the Lagrangian and Legendrian Gauss maps. We then show that a spectral parameter deformation of the Lagrangian harmonic Gauss map gives a harmonic map into the hyperbolic two-space for $-1< K<0$ or the two-sphere for $K>0$, respectively. Consequently, weakly complete constant Gaussian curvature surfaces with $-1 < K <0$ are in one-to-one correspondence with holomorphic quadratic differentials on the unit disk or the complex plane. |
| title | A classification of constant Gaussian curvature surfaces in the three-dimensional hyperbolic space |
| topic | Differential Geometry Primary 53C42, 58D10, Secondary 53C43, 53A10 |
| url | https://arxiv.org/abs/2404.08235 |