Embedded convex surfaces in hyperbolic and anti-de Sitter spaces

We show that given a quasi-circle $C$ in $\partial_{\infty}\mathbb{H}^3$ (respectively in $\partial_{\infty} \mathbb{ADS}^3$) and a complete conformal metric $h$ on $\mathbb{D}$ whose curvature $K_h$ takes values in a compact subset of $(-1,0)$ (respectively $(-\infty,-1)$), with all derivatives bounded with respect to the hyperbolic metric, there exists a smooth isometric embedding $V : (\mathbb{D}, h) \to \mathbb{H}^3$ (respectively $V : (\mathbb{D}, h) \to \mathbb{ADS}^3$) such that $V$ extends continuously to a homeomorphism $\partial V : \partial_{\infty}\mathbb{H}^2 \to C$. In the case of hyperbolic space, the statement still holds if $C$ is a Jordan curve.