  On the two dimensional sphere, we consider axisymmetric critical points of an
isoperimetric problem perturbed by a long-range interaction term. When the
parameter controlling the nonlocal term is sufficiently large, we prove the
existence of a local minimizer with arbitrary many interfaces in the
axisymmetric class of admissible functions. These local minimizers in this
restricted class are shown to be critical points in the broader sense (i.e.,
with respect to all perturbations). We then explore the rigidity, due to
curvature effects, in the criticality condition via several quantitative
results regarding the axisymmetric critical points.
