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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
MathSciNet review: 3199397
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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

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