On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface
On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface
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关于紧致黎曼曲面的无分支模空间的结构和维数的纯粹“黎曼”证明
DOI：
10.1007/bf01456093
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发表时间：
1984
影响因子：
1.4
通讯作者：
A. Tromba
中科院分区：
文献类型：
--
作者：
A. Fischer;A. Tromba
在上个世纪，Riemann给出了一个启启性的论证，即p属的紧曲面M的模空间（即固定格的共形不等价黎曼曲面的空间）依赖于6p-6个参数。1939年，在teichmijiller bbb的工作中给出了这一点的一般证明。Teichmtiller注意到，作为黎曼-洛克定理的一个结果，数字6p-6恰好是M.上的二次微分空间的维数。Teichmiiller也注意到H. Gr6tzsch的工作，他考虑了拟共形映射的概念。每个这样的映射都有一个与之相关的最大扩张（如[2,61]）。Gr6tzsch感兴趣的问题是在给定的映射类中最小化最大扩张。得到最小值的映射称为极值拟共形映射。
In the last century Riemann gave a heuristic argument that the moduli space (ie the space of conformally inequivalent Riemann surfaces of fixed genus) of a compact surface M of genus p> 1 depended on 6p-6 parameters. A general proof of this was given in 1939 in the work of TeichmiJller [21]. Teichmtiller noticed that as a consequence of the Riemann-Roch theorem the number 6p-6 was precisely the dimension of the space of quadratic differentials on M. Teichmiiller was also~ ware of the work of H. Gr6tzsch who considered the notion of quasi-conformal mappings. Each such mapping has a maximal dilation associated to it (eg see [2, 61]). Gr6tzsch was interested in the problem of minimizing the maximal dilation in a given class of mappings. A mapping for which the minimum is obtained is called extremal quasi-conformal.