A Survey of Scattering Theory on Riemann Surfaces with Applications in Global Analysis and Geometry

This paper gives an overview of our work on a scattering theory of one-forms and functions in a system of quasicircles on Riemann surfaces. It is rooted in an "overfare" process which takes a harmonic function on one side of the system of quasicircles to a harmonic function on the other side, with the same boundary values in a certain intrinsic non-tangential sense. This is bounded with respect to Dirichlet energy. If extra cohomological data is specified, one can apply this process to harmonic one-forms, and the resulting "scattering matrix" in terms of the holomorphic and anti-holomorphic components of the one-form is unitary. We describe applications to approximation theory, global analysis of singular integral operators on Riemann surfaces, and a new extension of the classical period map to surfaces of genus g with n boundary curves.