Ideal polyhedral surfaces in Fuchsian manifolds

TL;DR

This paper provides new variational proofs for the existence and uniqueness of ideal polyhedral surfaces in Fuchsian manifolds and for hyperbolic polyhedral metrics with prescribed curvature, enhancing understanding of hyperbolic geometry.

Contribution

It introduces a novel variational approach to establish the existence and uniqueness of ideal polyhedral surfaces and hyperbolic metrics in specified conformal classes.

Findings

Unique realization of boundary metrics as ideal Fuchsian polyhedra

New variational proof techniques for hyperbolic polyhedral metrics

Alternative proof of existence and uniqueness results

Abstract

Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a given conformal class.