Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties

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.