Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times\mathbb{R}$
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866913438113988608 |
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| author | Castro-Infantes, Jesús Manzano, José M. |
| author_facet | Castro-Infantes, Jesús Manzano, José M. |
| contents | For each $k\geq 3$, we construct a 1-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb{H}^2\times\mathbb{R}$ with genus $1$ and $k$ embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb{H}^2\times\mathbb{R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to $k$ vertical planes, as well as finite total curvature $-4kπ$. Finally, we also provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb{H}^2\times\mathbb{R}$ by the action of a hyperbolic or parabolic translation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_07028 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times\mathbb{R}$ Castro-Infantes, Jesús Manzano, José M. Differential Geometry Primary 53A10, Secondary 53C30 For each $k\geq 3$, we construct a 1-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb{H}^2\times\mathbb{R}$ with genus $1$ and $k$ embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb{H}^2\times\mathbb{R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to $k$ vertical planes, as well as finite total curvature $-4kπ$. Finally, we also provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb{H}^2\times\mathbb{R}$ by the action of a hyperbolic or parabolic translation. |
| title | Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times\mathbb{R}$ |
| topic | Differential Geometry Primary 53A10, Secondary 53C30 |
| url | https://arxiv.org/abs/2001.07028 |