= 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.
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