Parabolic and hyperbolic Riemann surfaces

Summary

An open Riemann surface is hyperbolic, in the sense of potential theory, when it carries a nonconstant negative subharmonic function, or equivalently a Green's function. It is parabolic when it is noncompact and carries no such function. This splits the open surfaces into those that are "large", like the disc, and those that are "small", like the plane. On a parabolic surface Brownian motion is recurrent, while on a hyperbolic surface it is transient.