Flat conformal structures on 3-manifolds. I: Uniformization of closed Seifert manifolds (Q688612)

This is the first in the series of three papers where we prove an existence theorem for flat conformal structures on finite-sheeted coverings over a wide class of Haken manifolds. In this paper we prove: Theorem 2.1. Let \(S(g,e)\) be a total space of a circle bundle over a closed orientable surface \(S_ g\) of a genus \(g\) having Euler number \(e \in \mathbb{Z}\) such that \(0<e \leq (g-1)/11\). Then \(S(g,e)\) admits an uniformizable flat conformal structure. Analogous statement was independently proven in the joint work by \textit{M. Gromov}, \textit{H. B. Lawson} and \textit{W. Thurston} [Publ. Math., Inst. Hautes Étud. Sci. 68, 27-45 (1988; Zbl 0692.57012)] and subsequent paper of \textit{N. Kuiper} [ibid. 47-76 (1988; Zbl 0692.57013)]. The theorem 2.1 is the first step in proving: Theorem. Let \(M\) be a closed Haken 3-manifold with unsolvable fundamental group such that the canonical composition of \(M\) from hyperbolic and Seifert components does not include gluing hyperbolic manifolds with hyperbolic or Euclidean ones. Then some finite-sheeted covering of \(M\) admits an uniformizable flat conformal structure. This result will be proven in the last of the series of papers. As an application of Theorem 2.1 we construct examples of uniformly quasiconformal actions of \(\mathbb{Z}_ n \times \pi_ 1(S_ g)\) which are not topologically equivalent to conformal actions. Another application: space of flat conformal structures on \(S(g,e)\) consists of at least \([(g-1)/11e]\) connected components.