  In this paper we collect some new observations about periodic critical points
and local minimizers of a nonlocal isoperimetric problem, arising in the
modeling of diblock copolymers. In the main result, by means of a purely
variational procedure, we show that it is possible to construct (locally
minimizing) periodic critical points whose shape resemble that of any given
strictly stable constant mean curvature (periodic) hypersurface. Along the way,
we establish several auxiliary results of independent interest.
