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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.17303 |
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| _version_ | 1866929729243709440 |
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| author | Assal, Fernando Al Lowe, Ben |
| author_facet | Assal, Fernando Al Lowe, Ben |
| contents | A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the 2-plane Grassmann bundle of M. When M has infinite volume and is geometrically finite, we show such sequences do not exist. As an application of the former, we obtain partial answers to the question of whether a negatively curved Riemannian 3-manifold that contains a sequence of asymptotically totally geodesic or totally umbilic surfaces must be hyperbolic. Finally, we give examples to show that if the dimension of M is greater than 3, the possible limiting behavior of asymptotically geodesic surfaces is less constrained than for totally geodesic surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17303 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically geodesic surfaces Assal, Fernando Al Lowe, Ben Differential Geometry Dynamical Systems Geometric Topology A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the 2-plane Grassmann bundle of M. When M has infinite volume and is geometrically finite, we show such sequences do not exist. As an application of the former, we obtain partial answers to the question of whether a negatively curved Riemannian 3-manifold that contains a sequence of asymptotically totally geodesic or totally umbilic surfaces must be hyperbolic. Finally, we give examples to show that if the dimension of M is greater than 3, the possible limiting behavior of asymptotically geodesic surfaces is less constrained than for totally geodesic surfaces. |
| title | Asymptotically geodesic surfaces |
| topic | Differential Geometry Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/2502.17303 |