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  1. 3 Open Mathematics

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  1. 3 Comerford M.

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  1. 1 2014

  2. 1 2013

  3. 1 2011

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Short separating geodesics for multiply connected domains

100%
Comerford M.
Open Mathematics
|
2011
|
tom 9
|
nr 5
984-996
EN
We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest geodesic always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a meridian of the domain. We prove that, although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible geodesics which separate the complement in this fashion.
2
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The Carathéodory topology for multiply connected domains II

100%
Comerford M.
Open Mathematics
|
2014
|
tom 12
|
nr 5
721-741
EN
We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.
3
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The Carathéodory topology for multiply connected domains I

100%
Comerford M.
Open Mathematics
|
2013
|
tom 11
|
nr 2
322-340
EN
We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.
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