This paper describes a procedure for the construction of monopoles on three-dimensional Euclideanspace, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rationalmaps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from$\Bbb{CP}^1$ to a suitable flag manifold. In describing the reverse direction, this paper completes the proofof the main theorem therein.

A construction of monopoles from solutions to Nahm's equations (a systemof ordinary differential equations) has been well-known for certain gauge groups for some time. Thesesolutions are hard to construct however, and the equations themselves become increasingly unwieldy when thegauge group is not $\mbox{SU}(2).$

Here, in contrast, a rational map is the only initial data. Butwhereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is onlyobtained from the rational map after solving a partial differential equation.

A non-linear flowequation, essentially just the path of steepest descent down the Yang-Mills-Higgs functional, is set up. Itis shown that, starting from an ‘approximate monopole’ - constructed explicitly from the rationalmap - a solution to the flow must exist, and converge to an exact monopole having the desired rationalmap.

1991 Mathematics Subject Classification: 53C07, 53C80, 58D27, 58E15, 58G11.