The induced metric and bending lamination on the boundary of convex hyperbolic 3-manifolds

Let $S$ be a closed hyperbolic surface and $M = \left ( 0,1 \right )$.
Suppose $h$ is a Riemannian metric on $S$ with curvature strictly greater than
$-1$, $h^{*}$ is a Riemannian metric on $S$ with curvature strictly less than
$1$, and every contractible closed geodesic with respect to $h^{*}$ has length
strictly greater than $2\pi$. Let $L$ be a measured lamination on $S$ such that
every closed leaf has weight strictly less than $\pi$. Then, we prove the
existence of a convex hyperbolic metric $g$ on the interior of $M$ that induces
the Riemannian metric $h$ (respectively $h^{*}$) as the first (respectively
third) fundamental form on $S \times \left\{ 0\right\}$ and induces a pleated
surface structure on $S \times \left\{ 1\right\}$ with bending lamination $L$.
This statement remains valid even in limiting cases where the curvature of $h$
is constant and equal to $-1$. Additionally, when considering a conformal class
$c$ on $S$, we show that there exists a convex hyperbolic metric $g$ on the
interior of $M$ that induces $c$ on $S \times \left\{ 0\right\}$, which is
viewed as one component of the ideal boundary at infinity of $(M,g)$, and
induces a pleated surface structure on $S \times \left\{ 0\right\}$ with
bending lamination $L$. Our proof differs from previous work by Lecuire for
these two last cases. Moreover, when we consider a lamination which is small
enough, in a sense that we will define, and a hyperbolic metric, we show that
the metric on the interior of M that realizes these data is unique.