Polyhedral surfaces in anti-de Sitter (2+1)-spacetimes
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909881546571776 |
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| author | Prosanov, Roman |
| author_facet | Prosanov, Roman |
| contents | We first prove that given a Fuchsian representation $ρ_\circ: π_1S \ra {\rm PSL}(2,\R)$, where $S$ is a closed oriented surface of genus $\geq 2$, any hyperbolic cone-metric on $S$ with cone-angles $>2π$ isometrically embeds as a future-convex bent Cauchy surface in a globally hyperbolic maximal Cauchy compact (GHMC) anti-de Sitter (2+1)-spacetime whose left representation is $ρ_\circ$. Second, we show that given any two such cone-metrics, there exists a GHMC anti-de Sitter (2+1)-spacetime in which the cone-metrics embed simultaneously, one as a future-convex bent Cauchy surface and one as a past-convex. Furthermore, in both cases we establish that such a spacetime and embeddings are unique provided that the cone-metrics are sufficiently small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polyhedral surfaces in anti-de Sitter (2+1)-spacetimes Prosanov, Roman Geometric Topology Differential Geometry Metric Geometry 53C50, 53C45, 53C24, 53C30, 57K35, 52A15 We first prove that given a Fuchsian representation $ρ_\circ: π_1S \ra {\rm PSL}(2,\R)$, where $S$ is a closed oriented surface of genus $\geq 2$, any hyperbolic cone-metric on $S$ with cone-angles $>2π$ isometrically embeds as a future-convex bent Cauchy surface in a globally hyperbolic maximal Cauchy compact (GHMC) anti-de Sitter (2+1)-spacetime whose left representation is $ρ_\circ$. Second, we show that given any two such cone-metrics, there exists a GHMC anti-de Sitter (2+1)-spacetime in which the cone-metrics embed simultaneously, one as a future-convex bent Cauchy surface and one as a past-convex. Furthermore, in both cases we establish that such a spacetime and embeddings are unique provided that the cone-metrics are sufficiently small. |
| title | Polyhedral surfaces in anti-de Sitter (2+1)-spacetimes |
| topic | Geometric Topology Differential Geometry Metric Geometry 53C50, 53C45, 53C24, 53C30, 57K35, 52A15 |
| url | https://arxiv.org/abs/2510.09313 |