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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Intrinsic circle domains
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by Edward Crane PDF
Conform. Geom. Dyn. 18 (2014), 65-84 Request permission

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
  • Received by editor(s): March 27, 2013
  • Published electronically: May 1, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 18 (2014), 65-84
  • MSC (2010): Primary 30C20; Secondary 30F45, 30C30, 52C26
  • DOI: https://doi.org/10.1090/S1088-4173-2014-00262-8
  • MathSciNet review: 3199397