  Flat magnetic nano-elements are an essential component of current and future
spintronic devices. By shaping an element it is possible to select and
stabilize chosen metastable magnetic states, control its magnetization
dynamics. Here, using a recent significant development in mathematics of
conformal mapping, complex variable based approach to the description of
magnetic states in planar nano-elements is extended to the case when elements
are multiply-connected (that is, contain holes or magnetic anti-dots). We show
that presence of holes implies a certain restriction on the set of magnetic
states of nano-element.

Mesoscale and Nanoscale Physics