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Conformal Geometry and Dynamics

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Faber and Grunsky operators corresponding to bordered Riemann surfaces


Author: Mohammad Shirazi
Journal: Conform. Geom. Dyn. 24 (2020), 177-201
MSC (2020): Primary 30F15; Secondary 30C35
DOI: https://doi.org/10.1090/ecgd/355
Published electronically: September 16, 2020
MathSciNet review: 4150224
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Abstract: Let $\mathfrak {R}$ be a compact Riemann surface of finite genus $\mathfrak {g}>0$ and let $\Sigma$ be the subsurface obtained by removing $n\geq 1$ simply connected regions $\Omega _1^+, \dots , \Omega _n^+$ from $\mathfrak {R}$ with non-overlapping closures. Fix a biholomorphism $f_k$ from the unit disc onto $\Omega _k^+$ for each $k$ and let $\mathbf {f}=(f_1, \dots , f_n)$. We assign a Faber and a Grunsky operator to $\mathfrak {R}$ and $\mathbf {f}$ when all the boundary curves of $\Sigma$ are quasicircles in $\mathfrak {R}$. We show that the Faber operator is a bounded isomorphism and the norm of the Grunsky operator is strictly less than one for this choice of boundary curves. A characterization of the pull-back of the holomorphic Dirichlet space of $\Sigma$ in terms of the graph of the Grunsky operator is provided.


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Additional Information

Mohammad Shirazi
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Canada
ORCID: 0000-0002-7311-6085
Email: shirazim@myumanitoba.ca, mohammad.shirazi@mail.mcgill.ca

Keywords: Faber operator, Grunsky operator, Schiffer operator, bordered Riemann surfaces, quasicircles
Received by editor(s): April 6, 2020
Received by editor(s) in revised form: July 21, 2020
Published electronically: September 16, 2020
Article copyright: © Copyright 2020 American Mathematical Society