Vortexes in planar ferromagnet and real Prym differentials

The distribution of the magnetization in a ferromagnetic is determined by many factors. In the case of a submicron scale planar magnetic the Heisenberg exchange interaction is dominant. Metastable states from a mathematical point of view are the harmonic maps of (multiply connected) domain to the sphere. D.Gross in 1978 proposed a series of local solutions with singularities modeling the magnetic vortexes. The general solution of this kind is associated with real Prym differentials that have a positive monodromy. In the report they will be explicitly listed.