Möbius structures on Seifert manifolds. I (Q1914750)

The paper studies geometrization of nontrivial 2-plane bundles \(W_{e,g}\) and \(S^1\)-bundles \(\partial W_{e,g}\) with a given Euler number \(e\) over a closed orientable surface \(S_g\) of genus \(g>1\). It is well-known that such \(S^1\)-bundles have \(\widetilde{PSL}_2\)-geometry, but the existence of conformal (=conformally flat) structures on them (correspondingly, complete hyperbolic metrics on 2-plane bundles \(W_{e,g}\)) is the main problem. First such hyperbolic and conformal structures on nontrivial 2-plane bundles \(W_{e,g}\) (on \(\partial W_{e,g}\)) were constructed by \textit{M. Gromov}, \textit{H. B. Lawson jun.} and \textit{W. Thurston} [Publ. Math., Inst. Hautes Etud. Sci. 68, 27-45 (1988; Zbl 0692.57012)], and the best known existence condition was due to N. Kuiper who showed that if \(|e|<2(g-1)/3\) then there exists a complete hyperbolic metric on \(W_{e,g}\). Studying conformal geometry on the products \(P\times S^1\) of the circle \(S^1\) and ``pairs of pants'' \(P\), the author constructively proves that \(|e|\leq g-1\) is a sufficient condition for existence of a complete hyperbolic metric on \(W_{e,g}\) (and of a conformally flat structure on the associated \(S^1\)-bundle \(\partial W_{e,g}\)).