Abstract
We consider some metrics and weak metrics defined on the Teichmüller 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 “ε 0-relative \({\epsilon}\)-thick parts”, and whose definition depends on the choice of some positive constants ε 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.
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Lixin Liu and Weixu Su were partially supported by NSFC (No. 10871211).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Liu, L., Papadopoulos, A., Su, W. et al. Length spectra and the Teichmüller metric for surfaces with boundary. Monatsh Math 161, 295–311 (2010). https://doi.org/10.1007/s00605-009-0145-8
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DOI: https://doi.org/10.1007/s00605-009-0145-8
Keywords
- Riemann surface with boundary
- Teichmüller space
- Teichmüller metric
- Length spectrum metric
- Length spectrum weak metrics
- Extremal length