On the hyperbolicity of certain planar domains (Q1093041)

Let \(S\) be a compact Riemann surface of genus greater than one and p: \(\tilde S\to S\) be an infinite covering. \textit{J. Dodziuk} [Acta Math. 152, 49-56 (1984; Zbl 0541.30035)] proved that \(\tilde S\) admits a nonzero \(L^ 2\) harmonic 1-form. The basic step in his proof is to demonstrate the existence of a nonconstant bounded subharmonic function on \(\tilde S\) when it is planar and he uses several deep results from differential geometry to establish this. This author gives a very short and simple proof of this result.