Embedded convex surfaces in hyperbolic and anti-de Sitter spaces

We show that given a quasi-circle $C$ in ${\mathrm{\partial }}_{\mathrm{\infty }}{\mathbb{H}}^3$ (respectively in ${\mathrm{\partial }}_{\mathrm{\infty }}{\mathbb{A}\mathbb{D}\mathbb{S}}^3$) and a complete conformal metric $h$ on $\mathbb{D}$ whose curvature $K_h$ takes values in a compact subset of $(-1,0)$ (respectively $(-\mathrm{\infty },-1)$), 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 $V:(\mathbb{D},h)\to {\mathbb{A}\mathbb{D}\mathbb{S}}^3$) such that $V$ extends continuously to a homeomorphism $\mathrm{\partial }V:{\mathrm{\partial }}_{\mathrm{\infty }}{\mathbb{H}}^2\to C$. In the case of hyperbolic space, the statement still holds if $C$ is a Jordan curve.