# The Carathéodory topology for multiply connected domains I

Open Mathematics (2013)

- Volume: 11, Issue: 2, page 322-340
- ISSN: 2391-5455

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topMark Comerford. "The Carathéodory topology for multiply connected domains I." Open Mathematics 11.2 (2013): 322-340. <http://eudml.org/doc/269062>.

@article{MarkComerford2013,

abstract = {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.},

author = {Mark Comerford},

journal = {Open Mathematics},

keywords = {Carathéodory topology; Hyperbolic geodesics; Meridians; hyperbolic geodesics; meridians},

language = {eng},

number = {2},

pages = {322-340},

title = {The Carathéodory topology for multiply connected domains I},

url = {http://eudml.org/doc/269062},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Mark Comerford

TI - The Carathéodory topology for multiply connected domains I

JO - Open Mathematics

PY - 2013

VL - 11

IS - 2

SP - 322

EP - 340

AB - 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.

LA - eng

KW - Carathéodory topology; Hyperbolic geodesics; Meridians; hyperbolic geodesics; meridians

UR - http://eudml.org/doc/269062

ER -

## References

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