  We characterize skyrmions in ultrathin ferromagnetic films as local
minimizers of a reduced micromagnetic energy appropriate for quasi
two-dimensional materials with perpendicular magnetic anisotropy and
interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out
in a suitable class of two-dimensional magnetization configurations that
prevents the energy from going to negative infinity, while not imposing any
restrictions on the spatial scale of the configuration. We first demonstrate
existence of minimizers for an explicit range of the model parameters when the
energy is dominated by the exchange energy. We then investigate the conformal
limit, in which only the exchange energy survives and identify the asymptotic
profiles of the skyrmions as degree 1 harmonic maps from the plane to the
sphere, together with their radii, angles and energies. A byproduct of our
analysis is a quantitative rigidity result for degree $\pm 1$ harmonic maps
from the two-dimensional sphere to itself.
