Real meromorphic differentials: a language for the meron configurations in planar nanomagnets

In this paper we use the language of real meromorphic differentials from the theory of Klein surfaces to describe the metastable states of multiply connected planar ferromagnetic nanoelements which minimize the exchange energy and have no side magnetic charges. Those solutions still have enough internal degrees of freedom which may serve as the Ritz parameters for minimization of further relevant energy terms or as the dynamical variables for the adiabatic approach. The nontrivial topology of the magnet itself brings us to several effects first described for the annulus and observed in the experiment. We explain the topological constraints on the numbers of vortexes and antivortexes in the magnet, as well as the algebraic constraints on their positions which stem from the Abel's theorem. The use of multivalued Prym differentials bring us to new meron configurations which were not considered in the seminal work of D.J.Gross.