A Riemann surface is said to be:
-Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function.
-Poincaré hyperbolic if it is covered by the unid disk.
Are this definitions equivalents? And why??
Potential theoretic hyperbolic implies poincare hyperbolic. This is just part of proof of uniformisation theorem. The other direction is false since any compact Riemann surface of genus at least two is poincare hyperbolic.