ISOPER;IMETRIC INEQUALITIES AND DIRICHLET FUNCTIONS OF

We prove that if a Riemann surface has a linear isoperimetric in-equality and verifies an extra condition of regularity, then ther e exists a non-constant harmonic function with finite Dirichlet inte-gral in the surface. We prove too, by an example, that the implication is not true without the condition of regularity. 1. Introduction. In this paper we study the relationship between linear isoperimetric inequalities and the existence of harmonic functions with finite Dirichle t integral on Riemann surfaces. By S we denote a Riemann surface (whose universal covering spac