On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space
On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space
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关于将拟共形Teichmüller空间纳入长谱Teichmüller空间
DOI：
10.1007/s00605-015-0813-9
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发表时间：
2012-01
期刊：
影响因子：
--
通讯作者：
Weixu Su
中科院分区：
文献类型：
--
作者：
D. Alessamdrini;刘立新;A. Papadopoulos;Weixu Su
本文是关于无限拓扑型曲面的。与有限类型曲面的情况不同，无限拓扑型曲面有几个变形空间。这种空间依赖于基点的选择(即固定保角结构或双曲结构的选择)，它们还依赖于标记的双曲结构的等价类集合上的距离的选择。我们讨论了两个形变空间，即拟共形Teichmüler空间和长度谱Teichmüler空间之间的比较问题。存在从拟共形空间到长度谱空间的自然包含映射，这并不总是满足性的。我们的工作假设是基点(双曲曲面)满足我们称为“上界”的条件。这意味着该曲面允许由长度在其上方有界的曲线定义的裤子分解。这一上界假设下的理论显示出一种两面性。一方面，有一些曲面满足我们所说的Shiga条件，即它们允许由长度上下有界的曲线定义的裤子分解。如果基点满足Shiga条件，则拟共形空间到长度谱空间的包含是满射的，并且是同胚的。在本文中，我们主要讨论另一类上界曲面，我们称之为“具有短内曲线的上界曲面”。这意味着相应的双曲曲面允许由长度在上面有界的曲线定义的裤子分解，并且使得一些内部曲线的长度接近于零。我们表明，在这种情况下，行为是完全不同的。在这个假设下，两个Teichmüler空间之间的包含映象在长度谱空间中是不稠密的。作为所用方法的推论，我们得到了长度谱Teichmüler空间在Fichel-Nielsen坐标下的显式参数化，并证明了长度谱Teichmüler空间是路径连通的。
This paper is about surfaces of infinite topological type. Unlike the case of surfaces of finite type, there are several deformation spaces associated with a surfaceSof infinite topological type. Such spaces depend on the choice of a basepoint (that is, the choice of a fixed conformal structure or hyperbolic structure onS) and they also depend on the choice of a distance on the set of equivalence classes of marked hyperbolic structures. We address the question of the comparison between two deformation spaces, namely, the quasiconformal Teichmüller space and the length-spectrum Teichmüller space. There is a natural inclusion map of the quasiconformal space into the length-spectrum space, which is not always surjective. We work under the hypothesis that the basepoint (a hyperbolic surface) satisfies a condition we call “upper-boundedness”. This means that this surface admits a pants decomposition defined by curves whose lengths are bounded above. The theory under this upper-boundedness hypothesis shows a dichotomy. On the one hand there are surfaces satisfying what we call Shiga’s condition, i.e. they admit a pants decomposition defined by curves whose lengths are bounded above and below. If the base point satisfies Shiga’s condition, then the inclusion of the quasiconformal space into the length-spectrum space is surjective, and it is a homeomorphism. In this paper we concentrate on the other kind of upper-bounded surfaces, which we call “upper-bounded with short interior curves”. This means that the corresponding hyperbolic surface admits a pants decomposition defined by curves whose lengths are bounded above, and such that the lengths of some interior curves approach zero. We show that in this case the behavior is completely different. Under this hypothesis, the image of the inclusion between the two Teichmüller spaces is nowhere dense in the length-spectrum space. As a corollary of the methods used, we obtain an explicit parametrization of the length-spectrum Teichmüller space in terms of Fenchel–Nielsen coordinates and we prove that the length-spectrum Teichmüller space is path-connected.
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DOI：
10.1017/s0305004114000656
发表时间：
2012-09
影响因子：
0.8
作者：
D. Šarić
通讯作者：
D. Šarić
影响因子：
0.6
作者：
Erina Kinjo
通讯作者：
Erina Kinjo
影响因子：
1.2
作者：
C. Bishop
通讯作者：
C. Bishop
DOI：
10.1215/s0012-7094-96-08310-6
发表时间：
1994-07
期刊：
--
影响因子：
--
作者：
Y. Minsky
通讯作者：
Y. Minsky
DOI：
--
发表时间：
2003
期刊：
--
影响因子：
--
作者：
Uller Space;H. Shiga
通讯作者：
Uller Space;H. Shiga