Extremal length and harmonic functions on Riemann surfaces

Abstract

Expressions for several conformally invariant pseudometrics on a Riemann surface R R are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on R R having finite Dirichlet integral. The reduced extremal distance between two points is defined on R R , on the Alexandroff one-point compactification of R R and on the Kerékjártó-Stoïlow compactification of R R . These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.