The Prescribed Metric on Convex Subsets of Anti-de Sitter Space with Quasi-Circle Ideal Boundaries

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 $(-\frac{1}{\epsilon },-1-\epsilon )$, for some $\epsilon >0$. Let $f$ be a quasi-symmetric map. We show the existence of a globally hyperbolic convex subset $\mathrm{\Omega }$ (see Definition 4.1) of the three-dimensional anti-de Sitter space, such that $\mathrm{\Omega }$ has $h^{+}$ (respectively $h^{-}$) as the induced metric on its future boundary (respectively on its past boundary) and has a gluing map ${\mathrm{\Phi }}_{\mathrm{\Omega }}$ (see Definition 5.7) equal to $f$.