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
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 ,
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.
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