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Relative Riemann mapping criteria and hyperbolic convexity


Author: Edward Crane
Journal: Proc. Amer. Math. Soc. 140 (2012), 2375-2382
MSC (2010): Primary 30C35, 30C62; Secondary 52A55
DOI: https://doi.org/10.1090/S0002-9939-2011-11096-7
Published electronically: October 27, 2011
MathSciNet review: 2898699
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Abstract: Let $ R$ be a simply-connected Riemann surface with a simply-connected subdomain $ U$. We give a criterion in terms of conformal reflections to determine whether $ R$ can be embedded in the complex plane so that $ U$ is mapped onto a disc. If it can, then $ U$ is convex with respect to the hyperbolic metric of $ R$, by a theorem of Jørgensen. We discuss the close relationship of our criterion to two generalizations of Jørgensen's theorem by Minda and Solynin. We generalize our criterion to the quasiconformal setting and also give a criterion for the multiply-connected case, where an embedding is sought that maps a given subdomain onto a circle domain.


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Additional Information

Edward Crane
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1UJ, United Kingdom

DOI: https://doi.org/10.1090/S0002-9939-2011-11096-7
Keywords: Riemann mapping, circle domains, hyperbolic convexity, conformal reflection, quasidiscs
Received by editor(s): November 12, 2010
Received by editor(s) in revised form: February 15, 2011
Published electronically: October 27, 2011
Additional Notes: This research was supported by the Heilbronn Institute for Mathematical Research
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.