Weakly almost-Fuchsian manifolds are nearly-Fuchsian
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912197622366208 |
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| author | Nguyen, Manh-Tien Schlenker, Jean-Marc Seppi, Andrea |
| author_facet | Nguyen, Manh-Tien Schlenker, Jean-Marc Seppi, Andrea |
| contents | We show that a hyperbolic three-manifold $M$ containing a closed minimal surface with principal curvatures in $[-1,1]$ also contains nearby (non-minimal) surfaces with principal curvatures in $(-1,1)$. When $M$ is complete and homeomorphic to $S\times\mathbb{R}$, for $S$ a closed surface, this implies that $M$ is quasi-Fuchsian, answering a question left open from Uhlenbeck's 1983 seminal paper. Additionally, our result implies that there exist (many) quasi-Fuchsian manifolds that contain a closed surface with principal curvatures in $(-1,1)$, but no closed minimal surface with principal curvatures in $(-1,1)$, disproving a conjecture from the 2000s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weakly almost-Fuchsian manifolds are nearly-Fuchsian Nguyen, Manh-Tien Schlenker, Jean-Marc Seppi, Andrea Differential Geometry We show that a hyperbolic three-manifold $M$ containing a closed minimal surface with principal curvatures in $[-1,1]$ also contains nearby (non-minimal) surfaces with principal curvatures in $(-1,1)$. When $M$ is complete and homeomorphic to $S\times\mathbb{R}$, for $S$ a closed surface, this implies that $M$ is quasi-Fuchsian, answering a question left open from Uhlenbeck's 1983 seminal paper. Additionally, our result implies that there exist (many) quasi-Fuchsian manifolds that contain a closed surface with principal curvatures in $(-1,1)$, but no closed minimal surface with principal curvatures in $(-1,1)$, disproving a conjecture from the 2000s. |
| title | Weakly almost-Fuchsian manifolds are nearly-Fuchsian |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2501.12277 |