This is the second in a series of four papers developing a scattering theory for harmonic one-forms on Riemann surfaces. In this paper we develop a conformally invariant characterization of the Sobolev space H^-1/2(Γ) where Γ is a border of a Riemann surface which is homeomorphic to the circle. We show that the boundary values of L² harmonic one-forms are in H^-1/2(Γ). We also consider the well-posedness of a similar Dirichlet problem on a Riemann surface with a finite number of borders homeomorphic to the circle. Furthermore, we prove an "overfare" result for a compact Riemann surface split into two surfaces by a complex of quasicircles.

The first author was partially supported by the National Sciences and Engineering Research Council of Canada. The second author is grateful to Andreas Strombergsson for partial financial support through a grant from Knut and Alice Wallenberg Foundation. The authors are also grateful to Rigund Staubach for preparing the figures in this manuscript. Finally we would like to thank the referee for suggestions which improved the presentation of the paper.