  The paper concerns the analysis of global minimizers of a Dirichlet-type
energy functional defined on the space of vector fields $H^1(S,T)$, where $S$
and $T$ are surfaces of revolution. The energy functional we consider is
closely related to a reduced model in the variational theory of micromagnetism
for the analysis of observable magnetization states in curved thin films. We
show that axially symmetric minimizers always exist, and if the target surface
$T$ is never flat, then any coexisting minimizer must have line symmetry. Thus,
the minimization problem reduces to the computation of an optimal
one-dimensional profile. We also provide a necessary and sufficient condition
for energy minimizers to be axially symmetric.
