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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

Intrinsic circle domains


Author: Edward Crane
Journal: Conform. Geom. Dyn. 18 (2014), 65-84
MSC (2010): Primary 30C20; Secondary 30F45, 30C30, 52C26
Published electronically: May 1, 2014
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Abstract: Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain $ \Omega $ in a compact Riemann surface $ S$. This means that each connected component $ B$ of $ S\setminus \Omega $ is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface $ (\Omega \cup B)$. Moreover, the pair $ (\Omega , S)$ is unique up to conformal isomorphisms. We give a generalization to countably infinite connectivity. Finally, we show how one can compute numerical approximations to intrinsic circle domains using circle packings and conformal welding.


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Additional Information

Edward Crane
Affiliation: Heilbronn Institute for Mathematical Research, School of Mathematics, University of Bristol, BS8 1TW, United Kingdom
Email: edward.crane@bristol.ac.uk

DOI: http://dx.doi.org/10.1090/S1088-4173-2014-00262-8
PII: S 1088-4173(2014)00262-8
Keywords: Circle domains, hyperbolic metric, circle packing, conformal welding
Received by editor(s): March 27, 2013
Published electronically: May 1, 2014
Article copyright: © Copyright 2014 American Mathematical Society