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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Shortening all the simple closed geodesics on surfaces with boundary


Authors: Athanase Papadopoulos and Guillaume Théret
Journal: Proc. Amer. Math. Soc. 138 (2010), 1775-1784
MSC (2000): Primary 32G15, 30F30, 30F60
Published electronically: December 28, 2009
MathSciNet review: 2587462
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple closed geodesics are shorter. (This is not possible for surfaces of finite type with empty boundary.) Furthermore, we show that we can do the shortening in such a way that it is bounded below by a positive constant. This improves a recent result obtained by Parlier. We include this result in a discussion of the weak metric theory of the Teichmüller space of surfaces with nonempty boundary.


References [Enhancements On Off] (What's this?)

  • 1. L. Liu, A. Papadopoulos, W. Su, G. Théret, On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary, preprint, arXiv:0903.0744v1, to appear in Annales Academiae Scientiarum Fennicae.
  • 2. Hugo Parlier, Lengths of geodesics on Riemann surfaces with boundary, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 2, 227–236. MR 2173363 (2006f:30050)
  • 3. W. Thurston, A spine for Teichmüller space, unpublished manuscript (1986).
  • 4. W. Thurston, Minimal stretch maps between hyperbolic surfaces, preprint, 1986, arXiv:math GT/9801039.

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Additional Information

Athanase Papadopoulos
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany – and – Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email: papadopoulos@math.u-strasbg.fr

Guillaume Théret
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Email: theret@mpim-bonn.mpg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10195-8
PII: S 0002-9939(09)10195-8
Keywords: Teichm\"uller space, surface with boundary, weak metric, length spectrum metric, Thurston's asymmetric metric.
Received by editor(s): March 27, 2009
Received by editor(s) in revised form: September 8, 2009
Published electronically: December 28, 2009
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.