Isoperimetric inequalities and Dirichlet functions of Riemann surfaces
Let \(\mathcal S\) be a complete Riemann surface with a metric of constant curvature \(-1\). The author shows that if \(\mathcal S\) has a linear isoperimetric inequality and also satisfies an extra condition of regularity, then there exists a non-constant harmonic function \(h\) with finite Dirichlet integral over \(\mathcal S\). The author gives an example of a surface \(\mathcal S\) where this regularity condition fails and where such an \(h\) does not exist.
scientific article; zbMATH DE number 722048