Title:

Magnetic monopoles and hyperbolic threemanifolds

Let M = H^{3}/Γ be a complete, noncompact, oriented geometrically finite hyperbolic 3manifold without cusps. By constructing a conformal compactification of M x S^{1} we functorially associate to M an oriented, conformally flat, compact 4manifold X (without boundary) with an S^{1}action. X determines M as a hyperbolic manifold. Using our functor and the differential geometry of conformally flat 4manifolds we prove that any Γ as above with a limit set of Hausdorff dimension ≤ 1 is Schottky, Fuchsian or extended Fuchsian. Furthermore, the Hodge theory for H^{2} (X;R) carries over to H^{1}(M, δM;R) and H^{2}(M;R) which correspond to the spaces of harmonic L^{2}forms of degree 1 and 2 on M. Comparison of lattices through the Hodge star gives an invariant h(M) ε GL(H^{2}(M;R)/GL(H^{2}(M;Z)) of the hyperbolic structure. Secondly we pay attention to magnetic monopoles on M which correspond to S^{1}invariant solutions of the antiselfduality equations on X. The basic result is that we associate to M an infinite collection of moduli spaces of monopoles , labelled by boundary conditions. We prove that the moduli spaces are not empty (under reasonable conditions), compute their dimension , prove orientability , the existence of a compactification and smoothness for generic S^{1}invariant conformal structures on X. For these results one doesn't need a hyperbolic structure on M , the existence of a conformal compactification X suffices. A twistor description for monopoles on a hyperbolic M can be given through the twistor space of X , and monopoles turn out to correspond to invariant holomorphic bundles on twistor space. We analyse these bundles. Explicit formulas for monopoles can be found on handlebodies M , and for M = surface x R we describe the moduli spaces in some detail.
