The prescribed metric on the boundary of convex subsets of anti-de Sitter space with a quasi-circle as ideal boundary

Let h⁺ and h⁻ be two complete, conformal metrics on the disc 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 (-1/ε, -1 - ε), for some ε > 0. Let f be a quasi-symmetric map. We show the existence of a globally hyperbolic convex subset Ω (see Definition 3.1) 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 ΦΩ (see Definition 4.5) equal to f.