### Abstract

For any multiply connected domain $\Omega$ in $\mathbb{R}^2$, let $S$ be the boundary of the convex hull in $H^3$ of $\mathbb{R}^2 \backslash \Omega $ which faces $\Omega$. Suppose in addition that there exists a lower bound $l > 0$ of the hyperbolic lengths of closed geodesics in $\Omega$. Then there is always a $K$-quasiconformal mapping from $S$ to $\Omega$, which extends continuously to the identity on $\partial S = \partial \Omega$, where $K$ depends only on $l$. We also give a numerical estimate of $K$ by using the parameter $l$.

Original language | English (US) |
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Pages (from-to) | 932-940 |

Number of pages | 9 |

Journal | Science China Mathematics |

Volume | 52 |

Issue number | 5 |

State | Published - 2009 |

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## Cite this

Liu, G., & Wu, S. (2009). Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant.

*Science China Mathematics*,*52*(5), 932-940.