Some counterexamples in the classification of open Riemann surfaces

Abstract

Recently Ahlfors and the author [1] constructed a Riemann surface of hyperbolic type which possessed no nonconstant harmonic functions with a finite Dirichlet integral. In the first section we explore some of the consequences of this example and construct a Riemann surface on which the spaces HD and HBD have dimension n. In the next section a bounded Riemann surface is exhibited which has no HD functions on it which vanish on the relative boundary, while it has a nonconstant HD whose normal derivative vanishes on the relative boundary. In the last section we use a refinement of the method in [1] to construct a Riemann surface admitting a nonconstant bounded harmonic function, but no nonconstant harmonic functions with a finite Dirichlet integral, thus demonstrating that the classes OHB and OHD are distinct.