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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Faber and Grunsky operators corresponding to bordered Riemann surfaces
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by Mohammad Shirazi
Conform. Geom. Dyn. 24 (2020), 177-201
Published electronically: September 16, 2020


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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Bibliographic Information
  • Mohammad Shirazi
  • Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Canada
  • ORCID: 0000-0002-7311-6085
  • Email:,
  • Received by editor(s): April 6, 2020
  • Received by editor(s) in revised form: July 21, 2020
  • Published electronically: September 16, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 24 (2020), 177-201
  • MSC (2020): Primary 30F15; Secondary 30C35
  • DOI:
  • MathSciNet review: 4150224