 

Eric Schippers and
Wolfgang Staubach
Overfare of harmonic oneforms on Riemann surfaces
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Published: 
October 5, 2024. 
Keywords: 
Overfare, scattering, Bordered Riemann surfaces, Quasicircles, Conformally nontangential limits, Conformal Sobolev spaces, L^{2} harmonic oneforms, Dirichlet problem, H^{1/2} Sobolev space. 
Subject [2020]: 
14F40, 30F15, 30F30, 35P99, 51M15. 


Abstract
This is the second in a series of four papers developing a scattering theory for harmonic oneforms on Riemann surfaces. In this paper we develop a conformally invariant characterization of the Sobolev space H^{1/2}(Γ) where Γ is a border of a Riemann surface which is homeomorphic to the circle. We show that the boundary values of L^{2} harmonic oneforms are in H^{1/2}(Γ). We also consider the wellposedness of a similar Dirichlet problem on a Riemann surface with a finite number of borders homeomorphic to the circle. Furthermore, we prove an "overfare" result for a
compact Riemann surface split into two surfaces by a complex of quasicircles.


Acknowledgements
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. The authors are also grateful to Rigund Staubach for preparing the figures in this manuscript. Finally we would like to thank the referee for suggestions which improved the presentation of the paper.


Author information
Eric Schippers
Machray Hall, Dept. of Mathematics
University of Manitoba
Winnipeg, MB R3T 2N2, Canada
eric.schippers@umanitoba.ca
Wolfgang Staubach
Department of Mathematics
Uppsala University
S751 06 Uppsala, Sweden
wulf@math.uu.se

