Bers' simultaneous uniformization and the intersection of Poincare holonomy varieties

TL;DR

This paper proves that the space of pairs of distinct complex projective structures with identical holonomy maps as a branched cover, reestablishes Bers' uniformization without quasi-conformal theory, and shows intersections of Poincaré holonomy varieties are discrete and non-empty.

Contribution

It provides a new proof of Bers' simultaneous uniformization theorem and establishes the discreteness and non-emptiness of intersections of Poincaré holonomy varieties.

Findings

The holomorphic map from the space of structures to Teichmüller space is a complete local branched cover.

The intersection of two Poincaré holonomy varieties is a non-empty discrete set.

Reproves Bers' uniformization theorem without quasi-conformal deformation theory.

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 Teichm\"uller 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 Poincar\'e holonomy varieties (${\rm SL}_2\mathbb{C}$-opers) is a non-empty discrete set, which is closely related to the mapping.