 Green function on a Riemann surface

With the positive logarithmic convention, a Green function with pole $p$ on a noncompact Riemann surface is positive and harmonic away from $p$, has local form $-\log |\zeta |$ plus a harmonic function in a coordinate $\zeta (p)=0$, and is minimal among positive functions with these properties. It is obtained, when finite, as the increasing limit of zero-boundary Green functions on an exhaustion by compact bordered subdomains. On the unit disk it isIts sign convention is $-\Delta G=2\pi {\delta }_p$ in a conformal coordinate, with the opposite overall sign from conventions using $\log |\zeta |$ as the singular part.