Approximation of infinite-dimensional Teichmüller spaces
Approximation of infinite-dimensional Teichmüller spaces
复制标题
无限维 Teichmüller 空间的近似
DOI：
10.1090/s0002-9947-1984-0728718-7
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发表时间：
1984
影响因子：
1.3
通讯作者：
F. Gardiner
中科院分区：
文献类型：
--
作者：
F. Gardiner
通过穷举过程，证明了无限维TcichMuller空间的TeichMuller度量与Kobavashi度量是等价的。利用同样的近似方法，将Reich·Strebel不等式的重要估计推广到无限维情形。这些估计被用来解释泰希穆勒度量如何是其无穷小形式的积分。它们还被用来给出一个序列是哈密尔顿泛函的绝对极大序列的充分条件。最后，利用它们给出了Beltrami系数序列收敛于Teichmuller度量的一个新的充分条件。导言。本文的主题是无限生成的Fuchsian群的Teichmiiller空间。通过涉及给定群的theta级数和有限生成子群的逼近技巧，我们将有限情形中已知的某些重要结果推广到无限生成情形。在§I中，我们建立了逼近技巧，并引用了涉及Poincare级数和有理函数逼近的必要定理。在§2中，我们考虑了具有复杂结构的Teichmiiller空间的Kobayashi极值问题，并证明了以下新结果。Royden关于Kobayashi和Teichmiiller Me的相等性的定理在无限情形下仍然成立。这些情形包括第一类和第二类群的Teichmiiller空间。特别是，泛Teichmiiller空间的情形也被包括在内。在§3中，我们证明了Reich和Strebel的重要的主要不等式[14]。它最显著的结果是对给定的Teichmiiller类的扩张的极值的上下限估计。这一节的主要结果是，即使在无限维的情况下，这些上限和下限估计也是成立的。在§4中，我们导出了众所周知的Teichmiiller度量的无穷小形式。结合Reich和Strebel[14]提出的哈密顿条件，我们证明了Teichmiiller度规等于其无穷小形式的积分。O‘Byrne在[6，7，13]中已经得到了这一结果。这里所用的方法更直接，证明的定理更具一般性。在§5中，我们给出了序列C(Jn)是哈密尔顿泛函H[p，)的绝对极大序列的一个充分条件。我们不知道编辑是否也收到了19R2和1月12日的情况。以修订后的形式。3月25日。1983年。19W数学学科分类。初级30C60；次级30C70。科尔……秩序和短语。泰奇穆勒空间。Teichmiillcr度量，小林度量。哈密尔顿泛函。绝对最大数列。无穷小度规。这项工作得到了纽约城市大学研究基金会的部分支持。367t·1984美国数学学会0002-9947/84$100+$0.25每页许可证或版权限制可能适用于再分发；请参阅http://www.ams.org/journal-terms-of-use
By means of an exhaustion process it is shown that Teichmuller's metric and Kobavashi's metric arc equal for infinite dimensional Tcichmuller spaces. By the same approximation method important estimates coming from the Reich·Strebel inequality are extended to the infinite dimensional cases. These estimates are used to ,how that Teichmuller's metric is the integral of its infinitesimal form. They are also used to give a sufficient condition for a sequence to be an absolute maximal sequence for the Hamilton functional. Finally, they are used to give a new sufficient condition for a sequence of Beltrami coefficients to converge in the Teichmuller metric. Introduction. The subject of this paper is Teichmiiller spaces of infinitely generated Fuchsian groups. By approximation techniques involving theta series and finitely generated subgroups of a given group, we extend certain important results already known in the finite case to the infinitely generated case. In § I we set up the approximation technique and cite the necessary theorems involving Poincare series and approximation by rational functions. In §2 we consider Kobayashi's extremal problem for Teichmiiller spaces with complex structure and prove the following new result. The theorem of Royden on the equality of the Kobayashi and Teichmiiller me tries remains true in the infinite cases. These cases include Teichmiiller spaces of groups of the first and second kind. In particular, the case of universal Teichmiiller space is included. In §3 we prove the important main inequality of Reich and Strebel [14). Its most significant consequences are upper and lower estimates for the extremal value of the dilatation in a given Teichmiiller class. The chief result of this section is that these upper and lower estimates hold even in the infinite dimensional cases. In §4 we derive the well-known [16) infinitesimal form of Teichmiiller's metric. We use this general form together with the Hamilton condition as developed by Reich and Strebel [14) to show that Teichmiiller's metric is equal to the integral of its infinitesimal form. O'Byrne already obtained this result in [6, 7, 13). The method used here is more direct and the theorem is proved in greater generality. In §5, we give a sufficient condition for a sequence C(Jn to be an absolute maximal sequence for the Hamilton functional H[p,). We do not know if this condition is also Received by the editors January 12, 19R2 and. in revised form. March 25. 1983. 19W Mathematics Suhject Classification. Primary 30C60; Secondary 30C70. Ker ..... ords and phrases. Teichmuller space. Teichmiillcr's metric, Kobayashi's metric. Hamilton functional. absolute maximal sequence. infinitesimal metric. I This work partially supported by the Research Foundation of CUNY. 367 t·1984 American Mathematical Society 0002-9947/84 $100 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use