This is the first in a series of four papers developing a scattering theory for harmonic functions/one-forms on Riemann surfaces. In this paper we prove the following. Let R be a compact Riemann surface split into two surfaces Σ₁ and Σ_2> by a complex of quasicircles. Given a harmonic function with L² derivatives on one of the pieces Σ₁, there is a unique harmonic function with L² derivatives on the other piece Σ₂ with the same boundary values as the original function in a certain conformally invariant non-tangential sense. We call the new harmonic function the overfare of the original function. This overfare map is well-defined and bounded with respect to Dirichlet semi-norm provided that Σ₁ is connected. For Weil-Petersson quasicircles, it is bounded with respect to the Sobolev H¹-norm.

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. Finally we would like to thank the referee for suggestions which improved the presentation of the paper.