Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities

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.Comment: 14 pages. We revised the old version greatly. In particular, we changed the title a little bit, generalized the main theorem to general Riemann surface, added a complex analytical definition for cone/cusp singularity of hyperbolic metric and Example 1.