 Hyperbolic Riemann surface in potential theory

A noncompact Riemann surface is hyperbolic in potential theory when its Green-function exhaustion has a finite limit, equivalently when it admits a positive Green function on a Riemann surface. This definition does not assume a disk uniformization. The unit disk is an example, since $-\log |z|$ is its positive Green function with pole zero. For simply connected surfaces this is equivalent to being conformally equivalent to the disk. For general surfaces, potential-theoretic hyperbolicity should not be conflated with the condition that the universal cover is the disk: some surfaces with disk universal cover have no positive Green function.