Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities

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 PSU(1, 1). As an application, we construct explicitly a new class of hyperbolic metrics with countably many singularities on the unit disc.