The geometric data on the boundary of convex subsets of hyperbolic manifolds

Let $N$ be a geodesically convex subset in a convex co-compact hyperbolic
manifold $M$ with incompressible boundary. We assume that each boundary
component of $N$ is either a boundary component of $\partial_\infty M$, or a
smooth, locally convex surface in $M$. We show that $N$ is uniquely determined
by the boundary data defined by the conformal structure on the boundary
components at infinity, and by either the induced metric or the third
fundamental form on the boundary components which are locally convex surfaces.
We also describe the possible boundary data. This provides an extension of both
the hyperbolic Weyl problem and the Ahlfors-Bers Theorem.
Using this statement for quasifuchsian manifolds, we obtain existence results
for similar questions for convex domains $\Omega\subset \HH^3$ which meets the
boundary at infinity $\partial_{\infty}\HH^3$ either along a quasicircle or
along a quasidisk. The boundary data then includes either the induced metric or
the third fundamental form in $\HH^3$, but also an additional ``gluing'' data
between different components of the boundary, either in $\HH^3$ or in
$\partial_\infty\HH^3$.