Eric Schippers: Overfare through curves in Riemann surfaces

Abstract:

We consider the following boundary value problem on a compact Riemann surface split into two pieces by a collection of Jordan curves. Given an \(L^2\) harmonic one-form on one of the pieces, we seek a harmonic one-form with the same boundary values and specified cohomology, which we call the "overfare" of the original form. When the Jordan curves are quasicircles, which are irregular curves arising in (for example) complex dynamics and Teichmüller theory, overfare is bounded. Furthermore, an associated scattering matrix is unitary. This has applications to approximation theory in complex analysis, Teichmüller theory, and conformal field theory. We give a non-technical overview of the analytic and geometric properties of overfare and related integral operators.