Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties

Abstract

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.