Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space

TL;DR

This paper proves that every complete finite area hyperbolic metric on a punctured sphere can be uniquely represented as the surface metric of a convex ideal polyhedron in hyperbolic 3-space, establishing a geometric correspondence.

Contribution

It establishes a unique realization of hyperbolic metrics on punctured spheres as convex ideal polyhedra in hyperbolic 3-space, linking surface metrics to 3D polyhedral geometry.

Findings

Unique realization of hyperbolic metrics as convex ideal polyhedra

Correspondence between surface metrics and 3D polyhedral geometry

Includes additional geometric observations

Abstract

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.