Hyperbolic manifolds with convex boundary

Let $(M,\mathrm{\partial }M)$ be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature $K>-1$, and that the third fundamental forms of $\text{dr}M$ are exactly the metrics with curvature $K<1$, for which contractible closed geodesics have length $L>2\pi$. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on $\text{dr}M$ is a linear combination of the first, second and third fundamental forms.