Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces

TL;DR

This paper proves the existence and uniqueness of convex polyhedral metrics with constant curvature and conical singularities on compact surfaces, generalizing classical results for spherical cases to broader settings.

Contribution

It establishes a comprehensive existence and uniqueness theorem for convex polyhedral metrics with constant curvature on compact surfaces, covering multiple cases including Lorentzian spaces.

Findings

Convex polyhedral metrics can be realized as convex polyhedra in space-forms.

Uniqueness holds up to global isometries within specified invariance conditions.

The theorem encompasses 10 distinct cases, extending classical spherical results.

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.