Let M = H<sup>3</sup>/Γ be a complete, non-compact, oriented geometrically finite hyperbolic 3-manifold without cusps. By constructing a conformal compactification of M x S<sup>1</sup> we functorially associate to M an oriented, conformally flat, compact 4-manifold X (without boundary) with an S<sup>1</sup>-action. X determines M as a hyperbolic manifold. Using our functor and the differential geometry of conformally flat 4-manifolds 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<sup>2</sup> (X;R) carries over to H<sup>1</sup>(M, δM;R) and H<sup>2</sup>(M;R) which correspond to the spaces of harmonic L<sup>2</sup>-forms of degree 1 and 2 on M. Comparison of lattices through the Hodge star gives an invariant h(M) ε GL(H<sup>2</sup>(M;R)/GL(H<sup>2</sup>(M;Z)) of the hyperbolic structure. Secondly we pay attention to magnetic monopoles on M which correspond to S<sup>1</sup>invariant solutions of the anti-self-duality 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<sup>1</sup>-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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:379883 |
| Date | January 1987 |
| Creators | Braam, Peter J. |
| Contributors | Atiyah, Michael Francis |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:daa73d43-6d58-404c-9926-ebf23f59cfc6 |
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