# Short separating geodesics for multiply connected domains

Open Mathematics (2011)

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

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topMark 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

top- [1] Ahlfors L.V., Complex Analysis, 3rd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1978
- [2] Buser P., Geometry and Spectra of Compact Riemann Surfaces, Progr. Math., 106, Birkhäuser, Boston, 1992 Zbl0770.53001
- [3] Buser P., Seppälä M., Short homology bases for Riemann surfaces, preprint available at http://www.math.fsu.edu/vseppala/papers/ShortHomology/ShortHomology.pdf Zbl1004.30030
- [4] Carleson L., Gamelin T.W., Complex Dynamics, Universitext Tracts Math., Springer, New York, 1993 Zbl0782.30022
- [5] Comerford M., A straightening theorem for non-autonomous iteration, preprint available at http://arxiv.org/abs/1106.4581
- [6] Conway J.B., Functions of One Complex Variable, Grad. Texts in Math., 11, Springer, New York-Heidelberg, 1973 Zbl0277.30001
- [7] Hubbard J.H., Teichmüller Theory and Applications to Geometry, Topology, and Dynamics - Volume 1: Teichmüller Theory, Matrix Editions, Ithaca, 2006 Zbl1102.30001
- [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] Milnor J., Dynamics in One Complex Variable, 3rd ed., Ann. of Math. Stud., 160, Princeton University Press, Princeton, 2006 Zbl1085.30002
- [10] Newman M.H.A., Elements of the Topology of Plane Sets of Points, 2nd ed., Cambridge University Press, Cambridge, 1961
- [11] Parlier H., The homology systole of hyperbolic Riemann surfaces, Geom. Dedicata (in press), DOI: 10.1007/s10711-011-9613-0 Zbl1246.53060
- [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