Abstract
AbstractWe 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.
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Analysis
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