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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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On the length spectrum Teichmüller spaces of Riemann surfaces of infinite type
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by Erina Kinjo PDF
Conform. Geom. Dyn. 22 (2018), 1-14 Request permission


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
  • MR Author ID: 942104
  • Email:
  • Received by editor(s): September 23, 2016
  • Received by editor(s) in revised form: October 2, 2017
  • Published electronically: February 26, 2018
  • © Copyright 2018 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 22 (2018), 1-14
  • MSC (2010): Primary 30F60; Secondary 32G15
  • DOI:
  • MathSciNet review: 3768110