Dual metrics on the boundary of strictly polyhedral hyperbolic 3-manifolds

TL;DR

This paper establishes a unique correspondence between certain spherical cone-metrics on the boundary of a hyperbolizable 3-manifold and strictly polyhedral hyperbolic metrics inside, under specific geometric conditions.

Contribution

It introduces a dual metric framework on the boundary of hyperbolic 3-manifolds and proves a uniqueness theorem for the associated polyhedral hyperbolic structures.

Findings

Existence of a unique strictly polyhedral hyperbolic metric for given boundary data.

Conditions on cone angles and geodesic lengths ensure metric uniqueness.

Extension of boundary metric theory to hyperbolic 3-manifolds with boundary.

Abstract

Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $>1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles are greater than $2\pi$ and the lengths of all closed geodesics that are contractible in $M$ are greater than $2\pi$ there exists a unique strictly polyhedral hyperbolic metric on $M$ such that $d$ is the induced dual metric on $\partial M$.