  Celebrated work of Alexandrov and Pogorelov determines exactly which metrics
on the sphere are induced on the boundary of a compact convex subset of
hyperbolic three-space. As a step toward a generalization for unbounded convex
subsets, we consider convex regions of hyperbolic three-space bounded by two
properly embedded disks which meet at infinity along a Jordan curve in the
ideal boundary. In this setting, it is natural to augment the notion of induced
metric on the boundary of the convex set to include a gluing map at infinity
which records how the asymptotic geometry of the two surfaces compares near
points of the limiting Jordan curve. Restricting further to the case in which
the induced metrics on the two bounding surfaces have constant curvature $K \in
[-1,0)$ and the Jordan curve at infinity is a quasicircle, the gluing map is
naturally a quasisymmetric homeomorphism of the circle. The main result is that
for each value of $K$, every quasisymmetric map is achieved as the gluing map
at infinity along some quasicircle. We also prove analogous results in the
setting of three-dimensional anti de Sitter geometry. Our results may be viewed
as universal versions of the conjectures of Thurston and Mess about prescribing
the induced metric on the boundary of the convex core of quasifuchsian
hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes.
