The classical Weyl problem asks whether any Riemannian metric of positive curvature on the sphere S2 can be uniquely realized as the induced metric on the boundary of a convex domain in Euclidean space. In hyperbolic space, there is an analogue which was solved by Alexandrov in the 1950s, but also a dual statement describing the possible third fundamental forms of the boundaries of bounded, convex domains.
We will describe those classical results, as well as some conjectural statements and partial results extending them either to convex domains in hyperbolic manifolds, or, more generally, to unbounded convex domains in H3.
Fanny Kassel
