  We carry out Monte Carlo simulations on fluid membranes with orientational
order and multiple edges in the presence and absence of external forces. The
membrane resists bending and has an edge tension, the orientational order
couples with the membrane surface normal through a cost for tilting, and there
is a chiral liquid crystalline interaction. In the absence of external forces,
a membrane initialized as a vesicle will form a disk at low chirality, with the
directors forming a smectic-A phase with alignment perpendicular to the
membrane surface except near the edge. At large chirality a catenoid-like shape
or a trinoid-like shape is formed, depending on the number of edges in the
initial vesicle. This shape change is accompanied by cholesteric ordering of
the directors and multiple $\pi$ walls connecting the membrane edges and
wrapping around the membrane neck. If the membrane is initialized instead in a
cylindrical shape and stretched by an external force, it maintains a nearly
cylindrical shape but additional liquid crystalline phases appear. For large
tilt coupling and low chirality, a smectic-A phase forms. For lower values of
the tilt coupling, a nematic phase appears at zero chirality with the average
director oriented perpendicular to the long axis of the membrane, while for
nonzero chirality a cholesteric phase appears. The $\pi$ walls are tilt walls
at low chirality and transition to twist walls as chirality is increased. We
construct a continuum model of the director field to explain this behavior.
