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On geodesics in asymptotic Teichmüller spaces

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Abstract

Let R be a Riemann surface of infinite analytic type, T(R) and AT(R) be the Teichmüller space and asymptotic Teichmüller space on R respectively. The purpose of this paper is to discuss some problems related to geodesics in AT(R). It is proved that uniqueness of geodesics joining two given points [μ] and [ν] in T (R) dose not imply uniqueness of geodesics joining [[μ]] and [[ν]] in AT(R). Furthermore, a Beltrami differential μ is constructed such that there are infinitely many geodesics joining [[0]] and [[μ]] in AT(R), and a sufficient condition to determine the difference of the geodesics [[tμ 1]] and [[tμ 2]] (0 ≤ t ≤ 1) joining [[0]] and [[μ]] in AT(Δ) is given.

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Authors and Affiliations

  1. Department of Applied Mathematics, Nanjing University of Science and Technology, 210094, Nanjing, People’s Republic of China

    Jinhua Fan

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  1. Jinhua Fan
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Correspondence to Jinhua Fan.

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This work is supported by Nanjing University of Science and Technology XKF09044.

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Fan, J. On geodesics in asymptotic Teichmüller spaces. Math. Z. 267, 767–779 (2011). https://doi.org/10.1007/s00209-009-0645-1

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  • DOI: https://doi.org/10.1007/s00209-009-0645-1

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