Abstract In our previous paper [23] it was proved that a sense-preserving homeomorphism g on the unit circle S 1 belongs to the Weil-Petersson class WP ( S 1 ) , namely, g can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is square integrable in the Poincare metric if and only if g is absolutely continuous such that log ⁡ g ′ belongs to the Sobolev class H 1 2 . In this sequel to [23] , we show that the smooth Hilbert manifold structure on WP ( S 1 ) inherited from H 1 2 by the pullback g ↦ log ⁡ | g ′ | is compatible with the standard Hilbert manifold structure introduced by Takhtajan-Teo [27] . This enables us to give a fast approach to some results in our previous papers [23] and [24] .

中文翻译：

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重新审视 Weil-Petersson Teichmüller 空间

摘要 在我们之前的论文 [23] 中，证明了单位圆 S 1 上的一个意义保持同胚 g 属于 Weil-Petersson 类 WP ( S 1 ) ，即 g 可以扩展为到单位圆的拟共形映射当且仅当 g 是绝对连续的，使得 log ⁡ g ′ 属于 Sobolev 类 H 1 2 时，其 Beltrami 系数是可平方积的圆盘。在 [23] 的续集中，我们展示了 WP ( S 1 ) 上的光滑希尔伯特流形结构通过回拉 g ↦ log ⁡ | 从 H 1 2 继承而来。g′ | 与 Takhtajan-Teo [27] 引入的标准 Hilbert 流形结构兼容。这使我们能够快速处理我们之前论文 [23] 和 [24] 中的一些结果。