The type problem and geometric structures on Riemann surfaces

Ara Basmajian
A Riemann surface 
of negative Euler characteristic has a dual existence both as a complex analytic object and as a geometric one. On the complex analytic side there is a long classical history of the study of the existence of particular types of functions on the Riemann surface. One such is the question of the existence or non-existence of a Green’s function. On the geometric side the unique hyperbolic metric associated to the Riemann surface carries with it a number of geometric invariants, and non-existence of a Green’s function is equivalent to the geodesic flow being ergodic. Moreover, for finite topological type surfaces it is well known that ergodicity of the geodesic flow is equivalent to the hyperbolic area being finite.
In this talk, after laying out some of the basics, we describe new re- sults involving the relationship between Fenchel-Nielsen coordinates and whether or not the surface carries a Green’s function on an infinite topo- logical type Riemann surface. This is joint work with Hrant Hakobyan and Dragomir Saric.