Spectral curves of non-integral hyperbolic monopoles

In (??), a is an SUn-connection on the trivial bundle over H, Fa is its curvature, iφ (the Higgs field) is a section of the adjoint bundle, and ∗ is the Hodge ∗-operator on H. We regard two monopoles as the same if they are gauge-equivalent. (The reason for our apparently eccentric notation for the Higgs field will become clear in §3.) We shall develop the twistor description of solutions of (??) from a somewhat novel point of view and then impose the boundary conditions that permit the spectral data of the monopole to be defined. An interesting consequence of the twistor description is a statement that, roughly speaking, hyperbolic monopoles are determined by their boundary data: this is reminiscent of one of the main results of [?] about certain SU2-monopoles, but the details are different. The motivation for this work comes from various sources. First of all, in view of the richness of the theory of Euclidean monopoles (an introduction to which can be found in the book of Atiyah and Hitchin [?]) it is to be expected that there is a similarly rich theory in the hyperbolic case. Indeed one may hope that the Euclidean theory could be recovered from the hyperbolic one by letting the curvature of H go to zero.