  Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly
convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$.
It was proved by Alexandrov that if $K$ is bounded, then it is uniquely
determined by the induced metric on the boundary, and any smooth metric with
curvature $K>-1$ can be obtained.
  We propose here an extension of the existence part of this result to
unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then
clearly not sufficient to determine $K$. However one can consider a richer data
on the boundary including the ideal boundary of $K$. Specifically, we consider
the data composed of the full conformal structure on the boundary of $K$ (in
the Poincar\'e model of $\HH^3$), together with the induced metric on $\partial
K$. We show that a wide range of "reasonable" data of this type, satisfying
mild curvature conditions, can be realized on the boundary of a convex subset
in $\HH^3$.

Differential Geometry;Geometric Topology