Shortening all the simple closed geodesics on surfaces with boundary

Athanase Papadopoulos (1, 2) , Guillaume Théret (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 MPI - Max-Planck-Institut für Mathematik

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the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.

Mots clés: Thurston's asymmetric metric.; weak metric; length spectrum metric; Thurston's asymmetric metric; Teichmüller space; surface with boundary

Domaines: Topologie géométrique [math.GT]

Athanase Papadopoulos, Guillaume Théret. Shortening all the simple closed geodesics on surfaces with boundary. Proceedings of the American Mathematical Society, 2010, 138, pp.1775-1784. ⟨10.1090/S0002-9939-09-10195-8⟩. ⟨hal-00389344v2⟩