Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities

TL;DR

This paper explores the relationship between hyperbolic metrics with isolated singularities and bounded projective functions on Riemann surfaces, providing explicit constructions of new metrics with countably many singularities.

Contribution

It establishes a correspondence linking hyperbolic metrics with isolated singularities to bounded projective functions with specific Schwarzian derivatives and monodromy conditions.

Findings

Constructed a new class of hyperbolic metrics with countably many singularities

Established a correspondence between metrics and projective functions with controlled Schwarzian derivatives

Provided explicit examples on the unit disc

Abstract

We establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in ${\rm PSU}(1,\,1)$. As an application, we construct explicitly a new class of hyperbolic metrics with countably many singularities on the unit disc.