The Prescribed Metric on Convex Subsets of Anti-de Sitter Space with Quasi-Circle Ideal Boundaries
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915561937567744 |
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| author | Mesbah, Abderrahim |
| author_facet | Mesbah, Abderrahim |
| contents | Let $h^{+}$ and $h^{-}$ be two complete, conformal metrics on the disc $\mathbb{D}$. Assume moreover that the derivatives of the conformal factors of the metrics $h^{+}$ and $h^{-}$ are bounded at any order with respect to the hyperbolic metric, and that the metrics have curvatures in the interval $\left(-\frac{1}ε, -1 - ε\right)$, for some $ε> 0$. Let $f$ be a quasi-symmetric map. We show the existence of a globally hyperbolic convex subset $Ω$ (see Definition 4.1) of the three-dimensional anti-de Sitter space, such that $Ω$ has $h^{+}$ (respectively $h^{-}$) as the induced metric on its future boundary (respectively on its past boundary) and has a gluing map $Φ_Ω$ (see Definition 5.7) equal to $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Prescribed Metric on Convex Subsets of Anti-de Sitter Space with Quasi-Circle Ideal Boundaries Mesbah, Abderrahim Differential Geometry Geometric Topology Let $h^{+}$ and $h^{-}$ be two complete, conformal metrics on the disc $\mathbb{D}$. Assume moreover that the derivatives of the conformal factors of the metrics $h^{+}$ and $h^{-}$ are bounded at any order with respect to the hyperbolic metric, and that the metrics have curvatures in the interval $\left(-\frac{1}ε, -1 - ε\right)$, for some $ε> 0$. Let $f$ be a quasi-symmetric map. We show the existence of a globally hyperbolic convex subset $Ω$ (see Definition 4.1) of the three-dimensional anti-de Sitter space, such that $Ω$ has $h^{+}$ (respectively $h^{-}$) as the induced metric on its future boundary (respectively on its past boundary) and has a gluing map $Φ_Ω$ (see Definition 5.7) equal to $f$. |
| title | The Prescribed Metric on Convex Subsets of Anti-de Sitter Space with Quasi-Circle Ideal Boundaries |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2407.08490 |