Moduli of punctured tori and the accessory parameter of Lamé’s equation
Moduli of punctured tori and the accessory parameter of Lamé’s equation
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穿孔环面模量及Lamé方程的附属参数
DOI：
10.1090/s0002-9947-1979-0542877-9
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发表时间：
1979
影响因子：
1.3
通讯作者：
A. Vasquez
中科院分区：
文献类型：
--
作者：
L. Keen;H. Rauch;A. Vasquez
为了解决黎曼曲面的均匀化和模化问题，必须构造覆盖空间和覆盖映射，并确定它们所依赖的参数。当黎曼曲面是被穿透的环面时，这可以通过几种方式非常明确地完成。覆盖映射由一个常微分方程--Lame方程联系起来。在这个方程中有一个常量，叫做“附件参数”。在本文中，我们从两个方面研究了这个辅助参数的行为。首先，我们利用Hill方法得到了不同均匀化的模与辅助参数之间的隐式关系。我们证明了附件参数不适合作为模数--即使是局部的。然后，我们使用计算机和数值技术来更明确地确定辅助参数的奇异性的性质。0。导言。在经典的环面均匀化理论中，模空间有两种截然不同的模型：上半平面的复解析T(周期比)和穿孔环面Fuchsian均匀化的Fricke实解析迹参数。我们最初的目标是找到TeichMuller空间的这些模型之间的显式联系。它们通过一个常微分方程式联系在一起，除了一个未知的常量外，这个常量是完全确定的，这个常量被称为“辅助参数”。附件参数的显式计算将给出所需的关系，并且似乎还给出了模空间的另一种模型。实际上，在现代版本[3]中，在穿孔环面的拟Fuchsian均匀化中出现的附加参数是模空间的复解析参数。附件参数本身就很有趣。在古典文学中对辅助参数有广泛的讨论，然而，具体的性质却知之甚少。我们希望我们在这里所做的工作对附件参数的一般问题有所帮助。在这篇文章中，我们证明了附加参数甚至不是TeichMuller空间的局部参数，并给出了关于它的定性信息；1978年5月17日被编辑收到。AMS(MOS)主题分类(1970)。初级30A58、30A46、30A60；次级32G15、30A48、14H15、20E40。这项研究得到了美国国家科学基金会拨款#mcs-77-02205和#mcs-77-02932的部分支持。
To solve the problems of uniformization and moduli for Riemann surfaces, covering spaces and covering mappings must be constructed, and the parameters on which they depend must be determined. When the Riemann surface is a punctured torus this can be done quite explicitly in several ways. The covering mappings are related by an ordinary differential equation, the Lame equation. There is a constant in this equation which is called the "accessory parameter". In this paper we study the behavior of this accessory parameter in two ways. First, we use Hill's method to obtain implicit relationships among the moduli of the different uniformizations and the accessory parameter. We prove that the accessory parameter is not suitable as a modulus-even locally. Then we use a computer and numerical techniques to determine more explicitly the character of the singularities of the accessory parameter. 0. Introduction. In the classical theory of uniformization of tori there are two distinct models for the space of moduli: the complex analytic T (the period ratio) in the upper half plane and the real analytic trace parameters of Fricke for the Fuchsian uniformization of a punctured torus. Our original goal was to find an explicit relation between these models of Teichmuller space. They are related via an ordinary differential equation which is completely determined except for an unknown constant, called the "accessory parameter". Explicit computation of the accessory parameter then would give the desired relationship and also seemed to give another model for the moduli space. Indeed, in a modem version [3], the accessory parameter which arises in the quasi-Fuchsian uniformization of the punctured torus is a complex analytic parameter for the moduli space. Accessory parameters are interesting in their own right. There is an extensive discussion of accessory parameters in the classical literature; however, little of a concrete nature was known. We hope that what we have done here sheds some light on the general problem of accessory parameters. In this paper, we prove that the accessory parameter is not even a local parameter for Teichmuller space and we give qualitative information about it; Received by the editors May 17, 1978. AMS (MOS) subject classifications (1970). Primary 30A58, 30A46, 30A60; Secondary 32G15, 30A48, 14H15, 20E40. 'This research was partially supported by NSF Grants #MCS-77-02205 and #MCS-77-02932. ? 1979 American Mathematical Society 0002-9947/79/0000-0508/$08.50
Whittaker 向量的空间不可约性
DOI：
--
发表时间：
2007
期刊：
影响因子：
--
作者：
Ikehata;M and Ohe;T;松本久義
通讯作者：
松本久義