Simultaneous Uniformization and Algebraic Correspondences

TL;DR

This paper generalizes Bers' simultaneous uniformization theorem to algebraic correspondences, constructing correspondences that uniformize pairs of non-homeomorphic genus zero orbifolds and linking Teichmüller space to correspondences.

Contribution

It introduces a novel algebraic approach to simultaneous uniformization, extending classical results to the setting of algebraic correspondences.

Findings

Constructed algebraic correspondences for non-homeomorphic genus zero orbifolds

Realized Teichmüller space of punctured spheres within correspondence space

Extended uniformization theory to algebraic correspondence framework

Abstract

We prove a generalization of Bers' simultaneous uniformization theorem in the world of algebraic correspondences. More precisely, we construct algebraic correspondences that simultaneously uniformize a pair of non-homeomorphic genus zero orbifolds. We also present a complex-analytic realization of the Teichm\"uller space of a punctured sphere in the space of correspondences.