  We define a gauged non-linear sigma model for a 2-sphere valued field and a
$SU(2)$ connection on an arbitrary Riemann surface whose energy functional
reduces to that for critically coupled magnetic skyrmions in the plane, with
arbitrary Dzyaloshinskii-Moriya interaction, for a suitably chosen gauge field.
We use the interplay of unitary and holomorphic structures to derive a general
solution of the first order Bogomol'nyi equation of the model for any given
connection. We illustrate this formula with examples, and also point out
applications to the study of impurities
