Hyperbolic manifolds with polyhedral boundary

Let (M, ∂M) be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. We are interested in the following question: Question A Let h be a (non-smooth) metric on ∂M , with curvature K > −1. Is there a unique hyperbolic metric g on M , with convex boundary, such that the induced metric on ∂M is h ? There is also a dual statement: Question B Let h be a (non-smooth) metric on ∂M , such that its universal cover is CAT(1). Is there a unique hyperbolic metric g on M , with convex boundary, such that the third fundamental form of ∂M is h ? Many partial results are known on those questions when M is a ball, and a few in more general cases. We are interested here in the special case where ∂M locally looks like an ideal polyhedron in H. We can give a fairly complete answer to question B — which in this case concerns the dihedral angles — and some partial results on question A. This also has some interesting by-products, for instance a flat affine structure on the Teichmüller space of a surface with some marked points.