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hal-00375691, version 1

## Length spectra and the Teichmueller metric for surfaces with boundary

Lixin Liu 1, Athanase Papadopoulos () 23, Weixu Su () 1, Guillaume Théret 3

Abstract: We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call $\varepsilon_0$-relative $\epsilon$-thick parts", and whose definition depends on the choice of some positive constants $\varepsilon_0$ and $\epsilon$. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.

• 1:  Department of Mathematics
• Zhongshan University
• 2:  Institut de Recherche Mathématique Avancée (IRMA)
• CNRS : UMR7501 – Université Louis Pasteur - Strasbourg I
• 3:  Max-Plank-Institut für Mathematik (MPI)
• Max-Planck-Institut
• Domain : Mathematics/Geometric Topology
• Keywords : Riemann surface with boundary – Teichmueller space – Teichmueller metric – length spectrum metric – ength spectrum weak metrics – extremal length.
• Available versions :  v1 (2009-04-15) v2 (2009-07-22)

• hal-00375691, version 1
• oai:hal.archives-ouvertes.fr:hal-00375691
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• Submitted on: Wednesday, 15 April 2009 17:49:44
• Updated on: Thursday, 16 April 2009 05:44:08