The Green's Functions of the Boundaries at Infinity of the Hyperbolic 3-Manifolds

TL;DR

This paper explores the extension of Manin's relation between Arakelov Green functions and geodesic configurations from single Riemann surfaces to more complex hyperbolic 3-manifolds with multiple boundary components, using various Kleinian groups.

Contribution

It provides partial results towards generalizing Manin's Green function relation to hyperbolic 3-manifolds with multiple boundary Riemann surfaces and different uniformization groups.

Findings

Partial generalizations of Manin's result achieved.

Connections established between Green functions and geodesic configurations.

Extensions to various Kleinian group uniformizations.

Abstract

The work is motivated by a result of Manin, which relates the Arakelov Green function on a compact Riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with Schottky uniformization, having the Riemann surface as conformal boundary at infinity. A natural question is to what extent the result of Manin can be generalized to cases where, instead of dealing with a single Riemann surface, one has several Riemann surfaces whose union is the boundary of a hyperbolic 3-manifold, uniformized no longer by a Schottky group, but by a Fuchsian, quasi-Fuchsian, or more general Kleinian group. We have considered this question in this work and obtained several partial results that contribute towards constructing an analog of Manin's result in this more general context.