Singular hyperbolic metrics and negative subharmonic functions

Abstract

We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is Zariski dense in $\text{PSL}(2,\,{\mathbb R})$. By using meromorphic differentials and affine connections, we obtain 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 $\text{PSL}(2,\,{\mathbb R})$. Moreover, we confirm the conjecture if the Riemann surface is the once punctured Riemann sphere, the twice punctured Riemann sphere, a once punctured torus or a compact Riemann surface.