  We consider a version of Gamow's liquid drop model with a short range
attractive perimeter-penalizing potential and a long-range Coulomb interaction
of a uniformly charged mass in $\R^3$. Here we constrain ourselves to
minimizing among the class of shapes that are columnar, i.e., constant in one
spatial direction. Using the standard perimeter in the energy would lead to
non-existence for any prescribed cross-sectional area due to the infinite mass
in the constant spatial direction. In order to heal this defect we use a
connected perimeter instead. We prove existence of minimizers for this
connected isoperimetric problem with long-range interaction and study the
shapes of minimizers in the small and large cross section regimes. For an
intermediate regime we use an Ohta-Kawasaki phase field model with
connectedness constraint to study the shapes of minimizers numerically.
