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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.14868 |
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| _version_ | 1866914692556914688 |
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| author | Smith, Graham |
| author_facet | Smith, Graham |
| contents | In the study of immersed surfaces of constant positive extrinsic curvature in space-forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such surfaces under the hypotheses of quasicompleteness. In particular, we show that, for $k>\text{Max}(0,-c)$, the only quasicomplete immersed surfaces of constant extrinsic curvature equal to $k$ in the $3$-dimensional space-form of constant sectional curvature equal to $c$ are the geodesic spheres. Together with earlier work of the author, this completes the classification of quasicomplete immersed surfaces of constant positive extrinsic curvature in $3$-dimensional space-forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14868 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On quasicomplete $k$-surfaces in $3$-dimensional space-forms Smith, Graham Differential Geometry 53A05 In the study of immersed surfaces of constant positive extrinsic curvature in space-forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such surfaces under the hypotheses of quasicompleteness. In particular, we show that, for $k>\text{Max}(0,-c)$, the only quasicomplete immersed surfaces of constant extrinsic curvature equal to $k$ in the $3$-dimensional space-form of constant sectional curvature equal to $c$ are the geodesic spheres. Together with earlier work of the author, this completes the classification of quasicomplete immersed surfaces of constant positive extrinsic curvature in $3$-dimensional space-forms. |
| title | On quasicomplete $k$-surfaces in $3$-dimensional space-forms |
| topic | Differential Geometry 53A05 |
| url | https://arxiv.org/abs/2211.14868 |