Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces

Abstract

We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover such a polyhedron is unique, up to global isometries, among convex polyhedra invariant under isometries acting on a totally umbilical surface. This general statement falls apart into 10 different cases. The cases when $S$ is the sphere are classical.

Survey paper. No proof. 10 pages

References in corpus (2)

• Rigidity of Polyhedral Surfaces
• Hyperideal polyhedra in hyperbolic manifolds