Ideal polyhedral surfaces in Fuchsian manifolds

Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a given conformal class.