The Weyl problem for unbounded convex domains in $\HH^3$

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 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
$\HH^3$.
We do not consider here the uniqueness of a convex subset with given boundary
data.