Polyhedral surfaces in anti-de Sitter (2+1)-spacetimes

Abstract

We first prove that given a Fuchsian representation $\rho_\circ: \pi_1S \ra {\rm PSL}(2,\R)$, where $S$ is a closed oriented surface of genus $\geq 2$, any hyperbolic cone-metric on $S$ with cone-angles $>2\pi$ 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 $\rho_\circ$. 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.