Relative Riemann mapping criteria and hyperbolic convexity
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- by Edward Crane PDF
- Proc. Amer. Math. Soc. 140 (2012), 2375-2382 Request permission
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.References
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Additional Information
- Edward Crane
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1UJ, United Kingdom
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2898699