Abstract:The existence of force-free magnetic fields in the exterior domain of some compact simply connected surface S is proved via an iteration scheme. The iteration starts with an arbitrary exterior vacuum field, which contains flux tubes originating and ending on S. At one cross-section of such a flux tube with S an arbitrary function is prescribed. For small values of (in the Hlder-norm 1, ; 0 1) the iteration is shown to converge to a force-free field with the prescribed values of in a flux tube which is close to the vacuum flux tube and 0 outside. The force-free field is close (in the Hlder- norm 1,) to the starting vacuum field, in particular, it has the same field line topology, the same boundary values on S and satisfies the same decay conditions in spatial infinity. It is in general three-dimensional and requires no continuous symmetries.