Singular Hyperbolic Metrics On Riemann Surfaces

This dissertation is a study of singular hyperbolic metrics on Riemann surfaces,which mainly contains two parts.On one hand,we give local models of hyperbolic metrics near isolated singularities using two methods.We prove that there exists a complex coordinate z centered at the singularity where the metric has the explicit expression of either.(4α²|z|^2α-2/(1-|z|^2α)²)|dz|² with α>0 or |z|^-2（In|z|）^-2|dz|².On the other hand,we propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface in the sense of the potential theory is Zariski dense in PSL（2,R）.We obtain some evidence of the conjecture that the monodromy group of the singular hyperbolic metric cannot be contained in four classes of one-dimensional Lie subgroups of PSL（2,R）.Moreover,we confirm the conjecture if the Riemann surface is a once punctured Riemann sphere,a twice punctured Riemann sphere,a once punctured torus or a compact Riemann surface.