Scattering of Harmonic Functions and Forms in Quasicircles
The Analysis and Geometry of Random Spaces March 28, 2022  April 01, 2022
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Riemann surfaces
quasicircles
Schiffer operators
harmonic oneforms
PlemeljSokhotski decomposition
Faber operator
Grunsky operator
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 seminorm characterizes quasicircles among Jordan curves. A similar process is obtained for L^2 harmonic oneforms by conjugating with differentiation. We extend this overfaring process for oneforms to collections of quasicircles separating a Riemann surface, and give an explicit expression in terms of integral operators of Schiffer related to the PlemeljSokhotski 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.
Scattering of Harmonic Functions and Forms in Quasicircles

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Scattering Of Harmonic Functions And Forms In Quasicircles
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