Horo-shrinkers in the hyperbolic space
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910322576588800 |
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| author | Bueno, Antonio López, Rafael |
| author_facet | Bueno, Antonio López, Rafael |
| contents | A surface $Σ$ in the hyperbolic space $\h^3$ is called a horo-shrinker if its mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle$, where $(x,y,z)$ are the coordinates of $\h^3$ in the upper half-space model and $N$ is the unit normal of $Σ$. In this paper we study horo-shrinkers invariant by one-parameter groups of isometries of $\h^3$ depending if these isometries are hyperbolic, parabolic or spherical. We characterize totally geodesic planes as the only horo-shrinkers invariant by a one-parameter group of hyperbolic translations. The grim reapers are defined as the horo-shrinkers invariant by a one-parameter group of parabolic translations. We describe the geometry of the grim reapers proving that they are periodic surfaces. In the last part of the paper, we give a complete classification of horo-shrinkers invariant by spherical rotations, distinguishing if the surfaces intersect or not the rotation axis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05527 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Horo-shrinkers in the hyperbolic space Bueno, Antonio López, Rafael Differential Geometry 53E10, 53C44, 53A10, 53C21, 53C42 A surface $Σ$ in the hyperbolic space $\h^3$ is called a horo-shrinker if its mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle$, where $(x,y,z)$ are the coordinates of $\h^3$ in the upper half-space model and $N$ is the unit normal of $Σ$. In this paper we study horo-shrinkers invariant by one-parameter groups of isometries of $\h^3$ depending if these isometries are hyperbolic, parabolic or spherical. We characterize totally geodesic planes as the only horo-shrinkers invariant by a one-parameter group of hyperbolic translations. The grim reapers are defined as the horo-shrinkers invariant by a one-parameter group of parabolic translations. We describe the geometry of the grim reapers proving that they are periodic surfaces. In the last part of the paper, we give a complete classification of horo-shrinkers invariant by spherical rotations, distinguishing if the surfaces intersect or not the rotation axis. |
| title | Horo-shrinkers in the hyperbolic space |
| topic | Differential Geometry 53E10, 53C44, 53A10, 53C21, 53C42 |
| url | https://arxiv.org/abs/2402.05527 |