It is known that every non-negatively curved metric on the 2-sphere admits a essentially unique realization as the intrinsic metric on the boundary of a convex body in the Euclidean 3-space. This story has two classical chapters: polyhedral and smooth, though there is also a common generalization obtained by Alexandrov and Pogorelov.
The work of Thurston from 70s highlighted the ubiquity and the diversity of hyperbolic manifolds among 3-dimensional ones. Hyperbolic 3-manifolds with convex boundary constitute a large and interesting class to study from various perspectives. In 90’s-00’s an analogue of the problem above for hyperbolic 3-manifolds with smooth strictly convex boundary was resolved in the works of Labourie and Schlenker. Curiously enough, a polyhedral counterpart was not known until recently. One of the reasons is that some metrics on the boundary of such 3-manifolds that are «intrinsically polyhedral» admit not so polyhedral realizations, which are somewhat more difficult to handle.
In my talk I will present a recent proof of the respective polyhedral result (the rigidity is shown in a generic case). Another outcome is a rigidity result for a family of convex cocompact hyperbolic 3-manifolds with respect to some geometric data on the boundaries of their convex cores. This is a step towards a resolution of conjectures of Thurston.