  Let M be a compact, orientable, mean convex 3-manifold with boundary. We show
that the set of all simple closed curves in the boundary of M which bound
unique area minimizing disks in M is dense in the space of simple closed curves
in the boundary of M which are nullhomotopic in M. We also show that the set of
all simple closed curves in the boundary of M which bound unique absolutely
area minimizing surfaces in M is dense in the space of simple closed curves in
the boundary of M which are nullhomologous in M.
