We consider the space of ordered pairs of distinct \({\mathbb{C}{\mathrm{P}}}^{1}\)-structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasi-Fuchsian space is identified with a connected component of this space. This space holomorphically maps to the product of the Teichmüller spaces minus its diagonal.

In this paper, we prove that this mapping is a complete local branched covering map. As a corollary, we reprove Bers’ simultaneous uniformization theorem without any quasi-conformal deformation theory. Our main theorem is that the intersection of arbitrary two Poincaré holonomy varieties (\(\operatorname{SL}_{2}\mathbb{C}\)-opers) is a non-empty discrete set, which is closely related to the mapping.

我们考虑黎曼曲面（任何方向）上具有相同完整性的不同\({\mathbb{C}{\mathrm{P}}}^{1}\)结构的有序对的空间，因此拟-Fuchsian 空间被标识为该空间的连通分量。该空间全纯映射到 Teichmüller 空间减去其对角线的乘积。

在本文中，我们证明该映射是一个完整的局部分支覆盖图。作为推论，我们在没有任何准共形变形理论的情况下反驳了 Bers 的同时均匀化定理。我们的主要定理是任意两个庞加莱完整簇 ( \(\operatorname{SL}_{2}\mathbb{C}\) -opers) 的交集是一个非空离散集，这与映射密切相关。