On geodesics in asymptotic Teichmüller spaces
On geodesics in asymptotic Teichmüller spaces
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DOI：
10.1007/s00209-009-0645-1
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发表时间：
2011-04
影响因子：
0.8
通讯作者：
Jinhua Fan
中科院分区：
文献类型：
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作者：
Jinhua Fan
设R是无穷解析型Riemann曲面，T（R）和AT（R）分别是R上的Teichmüller空间和渐近Teichmüller空间.本文的目的是讨论AT（R）中与测地线有关的一些问题。证明了T（R）中连接两个给定点[μ]和[ν]的测地线的唯一性并不意味着AT（R）中连接[[μ]和[[ν]]的测地线的唯一性。进而构造了一个Beltrami微分μ，使得在AT（R）中有无穷多条连接[[0]]和[[μ]]的测地线，并给出了确定连接[[0]]和[[μ]]的测地线[[tμ1]和[[tμ2]]（0 ≤t≤ 1）在AT（Δ）中的差的一个充分条件.
LetRbe a Riemann surface of infinite analytic type,T(R) andAT(R) be the Teichmüller space and asymptotic Teichmüller space onRrespectively. The purpose of this paper is to discuss some problems related to geodesics inAT(R). It is proved that uniqueness of geodesics joining two given points [μ] and [ν] inT(R) dose not imply uniqueness of geodesics joining [[μ]] and [[ν]] inAT(R). Furthermore, a Beltrami differentialμis constructed such that there are infinitely many geodesics joining [[0]] and [[μ]] inAT(R), and a sufficient condition to determine the difference of the geodesics [[tμ1]] and [[tμ2]] (0 ≤t≤ 1) joining [[0]] and [[μ]] inAT(Δ) is given.