  We investigate morphologies of semiflexible polymer rings, such as circular
DNA, which are adsorbed onto topographically or chemically structured substrate
surfaces. We classify all equilibrium morphologies for two striped surface
structures, (i) topographical surface grooves and (ii) chemically structured
surface domains. For both types of stripes, we find four equilibrium shapes: a
round toroidal and a confined elongated shape as well as two shapes containing
bulges. We determine the complete bifurcation diagram of these morphologies as
a function of their contour length and the ratio of adhesive strength to
bending rigidity. For more complex geometries consisting of several stripes we
find a cascade of transitions between elongated shapes. Finally, we compare our
findings to ring condensation by attractive interactions.
