A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866909262413824000 |
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| author | Castro-Infantes, Jesús Manzano, José M. Rodríguez, Magdalena |
| author_facet | Castro-Infantes, Jesús Manzano, José M. Rodríguez, Magdalena |
| contents | We show the existence of a $2$-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature $0\leq H\leq\frac{1}{2}$ in ${\mathbb{H}^2\times\mathbb{R}}$. They are symmetric with respect to a horizontal slice and a $k$ vertical planes disposed symmetrically, and extend the so called minimal saddle towers and $k$-noids. We show that the orientation plays a fundamental role when $H>0$ by analyzing their conjugate minimal surfaces in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ or $\mathrm{Nil}_3$. We also discover new complete examples that we call $(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders over curves of geodesic curvature $2H$ from the convex side, often giving rise to non-embedded examples if $H>0$.
In the discussion of embeddedness of the constructed examples, we prove that the Krust property does not hold for any $H>0$, i.e., there are minimal graphs over convex domains in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$, $\mathrm{Nil}_3$ or the Berger spheres, whose conjugate surfaces with constant mean curvature $H$ in $\mathbb{H}^2\times\mathbb{R}$ are not graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_13192 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property Castro-Infantes, Jesús Manzano, José M. Rodríguez, Magdalena Differential Geometry Primary 53A10, Secondary 53C30 We show the existence of a $2$-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature $0\leq H\leq\frac{1}{2}$ in ${\mathbb{H}^2\times\mathbb{R}}$. They are symmetric with respect to a horizontal slice and a $k$ vertical planes disposed symmetrically, and extend the so called minimal saddle towers and $k$-noids. We show that the orientation plays a fundamental role when $H>0$ by analyzing their conjugate minimal surfaces in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ or $\mathrm{Nil}_3$. We also discover new complete examples that we call $(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders over curves of geodesic curvature $2H$ from the convex side, often giving rise to non-embedded examples if $H>0$. In the discussion of embeddedness of the constructed examples, we prove that the Krust property does not hold for any $H>0$, i.e., there are minimal graphs over convex domains in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$, $\mathrm{Nil}_3$ or the Berger spheres, whose conjugate surfaces with constant mean curvature $H$ in $\mathbb{H}^2\times\mathbb{R}$ are not graphs. |
| title | A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property |
| topic | Differential Geometry Primary 53A10, Secondary 53C30 |
| url | https://arxiv.org/abs/2012.13192 |