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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Shortening all the simple closed geodesics on surfaces with boundary
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by Athanase Papadopoulos and Guillaume Théret PDF
Proc. Amer. Math. Soc. 138 (2010), 1775-1784 Request permission

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We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the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
  • 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.
  • Hugo Parlier, Lengths of geodesics on Riemann surfaces with boundary, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 2, 227–236. MR 2173363
  • W. Thurston, A spine for Teichmüller space, unpublished manuscript (1986).
  • 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
  • MR Author ID: 135835
  • 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
  • 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
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 1775-1784
  • MSC (2000): Primary 32G15, 30F30, 30F60
  • DOI: https://doi.org/10.1090/S0002-9939-09-10195-8
  • MathSciNet review: 2587462