A Weierstrass-type representation for harmonic maps from Riemann surfaces to general Lie groups.
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.
We study the relationship between linear isoperimetric inequalities and the existence of non-constant positive harmonic functions on Riemann surfaces.We also study the relationship between growth conditions of length of spheres and the existence and the existence of Green's function on Riemann surfaces.