Construction of Euclidian Monopoles

This paper describes a procedure for the construction of monopoleson three-dimensional Euclidean space, starting from their rationalmaps. A companion paper, ‘Euclidean monopoles and rationalmaps’, to appear in the same journal, describes the assignmentto a monopole of a rational map, from CP¹ to a suitable flagmanifold. In describing the reverse direction, this paper completesthe proof of the main theorem therein. A construction of monopoles from solutions to Nahm's equations(a system of ordinary differential equations) has been well-knownfor certain gauge groups for some time. These solutions arehard to construct however, and the equations themselves becomeincreasingly unwieldy when the gauge group is not SU(2). Here, in contrast, a rational map is the only initial data.But whereas one can be reasonably explicit in moving from Nahmdata to a monopole, here the monopole is only obtained fromthe rational map after solving a partial differential equation. A non-linear flow equation, essentially just the path of steepestdescent down the Yang-Mills-Higgs functional, is set up. Itis shown that, starting from an ‘approximate monopole’- constructed explicitly from the rational map - a solutionto the flow must exist, and converge to an exact monopole havingthe desired rational map. 1991 Mathematics Subject Classification:53C07, 53C80, 58D27, 58E15, 58G11.