Interpolation and Sampling for Generalized Bergman Spaces on finite Riemann surfaces.
Isoperimetric inequalities and Dirichlet functions of Riemann surfaces.
We prove that if a Riemann surface has a linear isoperimetric inequality and verifies an extra condition of regularity, then there exists a non-constant harmonic function with finite Dirichlet integral in the surface.We prove too, by an example, that the implication is not true without the condition of regularity.