The primary purpose of the paper is to study how a Riemann mapping depends on the corresponding Jordan curve. We are mainly concerned with those Jordan curves in the Weil-Petersson class, namely, the corresponding Riemann mappings can be quasiconformally extended to the whole plane with Beltrami coefficients being square integrable under the Poincare metric. We endow the space of all normalized Weil-Petersson curves with a new real Hilbert manifold structure and show that it is topologically equivalent to the standard complex Hilbert manifold structure.

中文翻译：

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Weil-Petersson Teichmüller 空间 III：Weil-Petersson 曲线的黎曼映射的相关性

本文的主要目的是研究黎曼映射如何依赖于相应的乔丹曲线。我们主要关注 Weil-Petersson 类中的那些 Jordan 曲线，即对应的黎曼映射可以拟共形地扩展到整个平面，其中贝尔特拉米系数在 Poincare 度量下是平方可积的。我们赋予所有归一化 Weil-Petersson 曲线的空间一个新的实 Hilbert 流形结构，并证明它在拓扑上等效于标准的复 Hilbert 流形结构。