Conjugate Plateau constructions in product spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866911962331348992 |
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| author | Castro-Infantes, Jesús Manzano, José M. Torralbo, Francisco |
| author_facet | Castro-Infantes, Jesús Manzano, José M. Torralbo, Francisco |
| contents | This survey paper investigates, from a purely geometric point of view, Daniel's isometric conjugation between minimal and constant mean curvature surfaces immersed in homogeneous Riemannian three-manifolds with isometry group of dimension four. On the one hand, we collect the results and strategies in the literature that have been developed so far to deal with the analysis of conjugate surfaces and their embeddedness. On the other hand, we revisit some constructions of constant mean curvature surfaces in the homogeneous product spaces $\mathbb{S}^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{R}^3$ having different topologies and geometric properties depending on the value of the mean curvature. Finally, we also provide some numerical pictures using Surface Evolver. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_13162 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Conjugate Plateau constructions in product spaces Castro-Infantes, Jesús Manzano, José M. Torralbo, Francisco Differential Geometry Primary 53A10, Secondary 53C30 This survey paper investigates, from a purely geometric point of view, Daniel's isometric conjugation between minimal and constant mean curvature surfaces immersed in homogeneous Riemannian three-manifolds with isometry group of dimension four. On the one hand, we collect the results and strategies in the literature that have been developed so far to deal with the analysis of conjugate surfaces and their embeddedness. On the other hand, we revisit some constructions of constant mean curvature surfaces in the homogeneous product spaces $\mathbb{S}^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{R}^3$ having different topologies and geometric properties depending on the value of the mean curvature. Finally, we also provide some numerical pictures using Surface Evolver. |
| title | Conjugate Plateau constructions in product spaces |
| topic | Differential Geometry Primary 53A10, Secondary 53C30 |
| url | https://arxiv.org/abs/2203.13162 |