On Teichmüller metric and the length spectrums of topologically infinite Riemann surfaces
On Teichmüller metric and the length spectrums of topologically infinite Riemann surfaces
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DOI：
10.2996/kmj/1309829545
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发表时间：
2011-06
影响因子：
0.6
通讯作者：
Erina Kinjo
中科院分区：
文献类型：
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作者：
Erina Kinjo
我们考虑由黎曼曲面的长度谱定义的Teichmüller空间T R 0上的度量d L. H.滋贺证明了d L在T上内斯了与Teichmüller度量d T相同的拓扑，如果一个黎曼曲面R 0可以被分解成两条裤子，使得它们的所有边界分量的长度（除了穿孔）从上到下都是一致有界的。在本文中，我们证明了存在无限型的黎曼曲面R 0，使得R 0不能分解成这样的裤子，而这两个度量定义了T <$R0 <$的相同拓扑。我们还给出了T_∞ R_0 ∞上这些度量有不同拓扑的充分条件，推广了Liu-Sun-Wei的一个结果.
We consider a metric d L on the Teichmu¨ller space T ð R 0 Þ defined by the length spectrum of Riemann surfaces. H. Shiga proved that d L defines the same topology as that of the Teichmu¨ller metric d T on T ð R 0 Þ if a Riemann surface R 0 can be decomposed into pairs of pants such that the lengths of all their boundary components except punctures are uniformly bounded from above and below. In this paper, we show that there exists a Riemann surface R 0 of infinite type such that R 0 cannot be decomposed into such pairs of pants, whereas the two metrics define the same topology on T ð R 0 Þ . We also give a su‰cient condition for these metrics to have di¤erent topologies on T ð R 0 Þ , which is a generalization of a result given by Liu-Sun-Wei.