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A symplectic functional analytic proof of the conformal welding theorem


Authors: Eric Schippers and Wolfgang Staubach
Journal: Proc. Amer. Math. Soc. 143 (2015), 265-278
MSC (2010): Primary 30C35, 30C62, 30F60; Secondary 53D30
DOI: https://doi.org/10.1090/S0002-9939-2014-12225-8
Published electronically: September 24, 2014
MathSciNet review: 3272752
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Abstract: We give a new functional-analytic/symplectic geometric proof of the conformal welding theorem. This is accomplished by representing composition by a quasisymmetric map $ \phi $ as an operator on a suitable Hilbert space and algebraically solving the conformal welding equation for the unknown maps $ f$ and $ g$ satisfying $ g \circ \phi = f$. The univalence and quasiconformal extendibility of $ f$ and $ g$ is demonstrated through the use of the Grunsky matrix.


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Additional Information

Eric Schippers
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada
Email: eric_schippers@umanitoba.ca

Wolfgang Staubach
Affiliation: Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
Email: wulf@math.uu.se

DOI: https://doi.org/10.1090/S0002-9939-2014-12225-8
Keywords: Conformal welding, Grunsky matrix, infinite Siegel disk, quasi-symmetries, conformal maps
Received by editor(s): February 25, 2013
Received by editor(s) in revised form: March 27, 2013
Published electronically: September 24, 2014
Additional Notes: The first author was partially supported by the National Sciences and Engineering Research Council.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.