### Abstract

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain U in a compact Riemann surface S. This means that each connected component B of S\U is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface U union B. Moreover, the pair (U, 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.

Original language | English |
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Pages (from-to) | 65-84 |

Number of pages | 20 |

Journal | Conformal Geometry and Dynamics |

Volume | 18 |

Early online date | 1 May 2014 |

DOIs | |

Publication status | Published - 1 May 2014 |

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### Keywords

- circle domains
- hyperbolic metric
- circle packing
- conformal welding