Short separating geodesics for multiply connected domains

Mark Comerford

Open Mathematics (2011)

  • Volume: 9, Issue: 5, page 984-996
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Mark Comerford. "Short separating geodesics for multiply connected domains." Open Mathematics 9.5 (2011): 984-996. <http://eudml.org/doc/269176>.

@article{MarkComerford2011,
abstract = {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.},
author = {Mark Comerford},
journal = {Open Mathematics},
keywords = {Hyperbolic Geodesics; Meridians; hyperbolic geodesics; meridians; multiply connected domains},
language = {eng},
number = {5},
pages = {984-996},
title = {Short separating geodesics for multiply connected domains},
url = {http://eudml.org/doc/269176},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Mark Comerford
TI - Short separating geodesics for multiply connected domains
JO - Open Mathematics
PY - 2011
VL - 9
IS - 5
SP - 984
EP - 996
AB - 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.
LA - eng
KW - Hyperbolic Geodesics; Meridians; hyperbolic geodesics; meridians; multiply connected domains
UR - http://eudml.org/doc/269176
ER -

References

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  8. [8] Keen L., Lakic N., Hyperbolic Geometry from a Local Viewpoint, London Math. Soc. Stud. Texts, 68, Cambridge University Press, Cambridge, 2007 http://dx.doi.org/10.1017/CBO9780511618789 Zbl1190.30001
  9. [9] Milnor J., Dynamics in One Complex Variable, 3rd ed., Ann. of Math. Stud., 160, Princeton University Press, Princeton, 2006 Zbl1085.30002
  10. [10] Newman M.H.A., Elements of the Topology of Plane Sets of Points, 2nd ed., Cambridge University Press, Cambridge, 1961 
  11. [11] Parlier H., The homology systole of hyperbolic Riemann surfaces, Geom. Dedicata (in press), DOI: 10.1007/s10711-011-9613-0 Zbl1246.53060
  12. [12] Parlier H., Separating simple closed geodesics and short homology bases on Riemann surfaces, preprint available at http://homeweb.unifr.ch/parlierh/pub/articleHomology.pdf 

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