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hal-00389344, version 2

Shortening all the simple closed geodesics on surfaces with boundary

Athanase Papadopoulos () 1, 2, Guillaume Théret 2

Abstract: We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the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 in [2]. We include this result in a discussion of the weak metric theory of the Teichmüller space of surfaces with nonempty boundary.

  • 1:  Institut de Recherche Mathématique Avancée (IRMA)
  • CNRS : UMR7501 – Université Louis Pasteur - Strasbourg I
  • 2:  Max-Plank-Institut für Mathematik (MPI)
  • Max-Planck-Institut
  • Domain : Mathematics/Geometric Topology
  • Keywords : Teichmüller space – surface with boundary – weak metric – length spectrum metric – Thurston's asymmetric metric.
  • Comment : Revised version – to appear in the Proceedings of the AMS.
  • Available versions :  v1 (2009-05-28) v2 (2009-09-09)
 
  • hal-00389344, version 2
  • http://hal.archives-ouvertes.fr/hal-00389344
  • oai:hal.archives-ouvertes.fr:hal-00389344
  • From: Athanase Papadopoulos <>
  • Submitted on: Wednesday, 9 September 2009 11:42:09
  • Updated on: Wednesday, 9 September 2009 11:44:23