Generalized convexity and quantitative estimates for constant mean curvature spacelike hypersurfaces in Anti-de Sitter space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915433746006016 |
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| author | Trebeschi, Enrico |
| author_facet | Trebeschi, Enrico |
| contents | We study the principal curvatures of properly embedded constant mean curvature hypersurfaces in the Anti-de Sitter space $\mathbb{H}^{n,1}$. We generalize the notion of convex hull and give an upper bound on the principal curvatures which only depends on the width of the $H-$shifted convex hull. This analysis has two direct consequences. First, it allows to bound the sectional curvature of $H-$hypersurfaces by an explicit function of the the width of the $H-$shifted convex hull. Second, we bound the quasiconfromal dilatation of a class of quasiconformal maps on the hyperbolic plane $\mathbb{H}^2$, called $θ-$landslides, in terms of the cross-ratio norm of their quasi-symmetric extension on $\partial_\infty\mathbb{H}^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05349 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized convexity and quantitative estimates for constant mean curvature spacelike hypersurfaces in Anti-de Sitter space Trebeschi, Enrico Differential Geometry We study the principal curvatures of properly embedded constant mean curvature hypersurfaces in the Anti-de Sitter space $\mathbb{H}^{n,1}$. We generalize the notion of convex hull and give an upper bound on the principal curvatures which only depends on the width of the $H-$shifted convex hull. This analysis has two direct consequences. First, it allows to bound the sectional curvature of $H-$hypersurfaces by an explicit function of the the width of the $H-$shifted convex hull. Second, we bound the quasiconfromal dilatation of a class of quasiconformal maps on the hyperbolic plane $\mathbb{H}^2$, called $θ-$landslides, in terms of the cross-ratio norm of their quasi-symmetric extension on $\partial_\infty\mathbb{H}^2$. |
| title | Generalized convexity and quantitative estimates for constant mean curvature spacelike hypersurfaces in Anti-de Sitter space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.05349 |