Bers' simultaneous uniformization and the intersection of Poincare holonomy varieties

Abstract

We consider the space of ordered pairs of distinct $\mathbb{C}{P}^1$-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 (${\mathrm{SL}}_2\mathbb{C}$-opers) is a non-empty discrete set, which is closely related to the mapping.

Dedicated to Misha Kapovich on the occasion of his 60th birthday. 71 pages, 26 figures. To appear in GAFA

References in corpus (1)

• Uniformization of compact complex manifolds by Anosov representations