Scattering of Harmonic Functions and Forms in Quasicircles

Given a harmonic function h of bounded Dirichlet energy on a quasidisk, there exists a harmonic function of bounded Dirichlet energy on the interior of its complement with the same boundary values (except on a negligible set). We call this the overfare of h. The existence of a bounded overfare with respect to the Dirichlet semi-norm characterizes quasicircles among Jordan curves. A similar process is obtained for L^2 harmonic one-forms by conjugating with differentiation. We extend this overfaring process for one-forms to collections of quasicircles separating a Riemann surface, and give an explicit expression in terms of integral operators of Schiffer related to the Plemelj-Sokhotski jump decomposition. We use this to elucidate the geometric and analytic meaning of the Grunsky and Faber operators and their generalizations. This process is further related to the geometry of the specific surface and moduli space, for example providing index theorems for the Schiffer operators. If time allows we also discuss analogies with scattering theory. Joint work with Wulf Staubach.

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