On local comparison between various metrics on Teichmüller spaces
On local comparison between various metrics on Teichmüller spaces
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DOI：
10.1007/s10711-011-9601-4
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发表时间：
2010-12
影响因子：
0.5
通讯作者：
D. Alessandrini;Li-Xing Liu;A. Papadopoulos;W. Su
中科院分区：
文献类型：
--
作者：
D. Alessandrini;Li-Xing Liu;A. Papadopoulos;W. Su
在选择了特定的基点(曲面上的复数或双曲结构)后，无限拓扑型曲面上有几个Teichmüler空间。这类空间包括拟共形Teichmüler空间、长度谱Teichmüler空间、Fancel-Nielsen Teichmüler空间，以及其他空间。一般来说，这些空间是集合的--理论上是不同的。因此，一个重要的问题是理解它们之间的关系。每个空间都配备了自己的度量，在某些假设下，它们之间存在包含。在本文中，我们得到了这些包含的局部度量比较结果，即在一定的假设下，我们证明了这些包含是局部双Lipschitz的。为了得到这些结果，我们使用了一些双曲几何估计，这些估计也给出了有限类型曲面的新结果。我们记得，对于有限类型的曲面，所有这些Teichmüler空间在集合方向上是重合的。在没有边界分支(但可能有穿孔)的有限类型曲面的情形下，我们证明了单位映射到Teichmüler空间的任意厚部分的限制关于区域上的长度谱度量和值域上的经典Teichmüler度量是全局双Lipschitz的。对于具有穿孔和边界分量的有限类型曲面，在Teichmüler空间上有一种度量，我们称之为弧度量，它的定义类似于长度谱度量，但它使用测地弧的长度而不是闭测地线的长度。我们证明了单位映射对Teichmüler空间中任何“相对厚”部分的限制是全局双Lipschitz的，它关于定义域和值域上的长度谱度量、Teichmüler度量和弧度量中的任一个。
There are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). Such spaces include the quasiconformal Teichmüller space, the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller space, and there are others. In general, these spaces are set-theoretically different. An important question is therefore to understand relations between them. Each of these spaces is equipped with its own metric, and under some hypotheses, there are inclusions between them. In this paper, we obtain local metric comparison results on these inclusions, namely, we show that the inclusions are locally bi-Lipschitz under certain hypotheses. To obtain these results, we use some hyperbolic geometry estimates that give new results also for surfaces of finite type. We recall that in the case of a surface of finite type, all these Teichmüller spaces coincide setwise. In the case of a surface of finite type with no boundary components (but possibly with punctures), we show that the restriction of the identity map to any thick part of Teichmüller space is globally bi-Lipschitz with respect to the length spectrum metric on the domain and the classical Teichmüller metric on the range. In the case of a surface of finite type with punctures and boundary components, there is a metric on the Teichmüller space which we call the arc metric, whose definition is analogous to the length spectrum metric, but which uses lengths of geodesic arcs instead of lengths of closed geodesics. We show that the restriction of the identity map to any “relative thick” part of Teichmüller space is globally bi-Lipschitz, with respect to any of the three metrics: the length spectrum metric, the Teichmüller metric and the arc metric on the domain and on the range.