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On the length spectrum Teichmüller spaces of Riemann surfaces of infinite type


Author: Erina Kinjo
Journal: Conform. Geom. Dyn. 22 (2018), 1-14
MSC (2010): Primary 30F60; Secondary 32G15
DOI: https://doi.org/10.1090/ecgd/316
Published electronically: February 26, 2018
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Abstract: On the Teichmüller space $ T(R_0)$ of a hyperbolic Riemann surface $ R_0$, we consider the length spectrum metric $ d_L$, which measures the difference of hyperbolic structures of Riemann surfaces. It is known that if $ R_0$ is of finite type, then $ d_L$ defines the same topology as that of Teichmüller metric $ d_T$ on $ T(R_0)$. In 2003, H. Shiga extended the discussion to the Teichmüller spaces of Riemann surfaces of infinite type and proved that the two metrics define the same topology on $ T(R_0)$ if $ R_0$ satisfies some geometric condition. After that, Alessandrini-Liu-Papadopoulos-Su proved that for the Riemann surface satisfying Shiga's condition, the identity map between the two metric spaces is locally bi-Lipschitz.

In this paper, we extend their results; that is, we show that if $ R_0$ has bounded geometry, then the identity map $ (T(R_0),d_L) \to (T(R_0),d_T)$ is locally bi-Lipschitz.


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Additional Information

Erina Kinjo
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama 2-12-1, Meguro-ku, Tokyo 152-8551, Japan
Email: kinjo.e.aa@m.titech.ac.jp

DOI: https://doi.org/10.1090/ecgd/316
Keywords: Length spectrum, Teichm\"uller metric, Riemann surface of infinite type
Received by editor(s): September 23, 2016
Received by editor(s) in revised form: October 2, 2017
Published electronically: February 26, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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