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