  We predict that a low-permittivity oblate body (disk-shaped object) above a
thin metal substrate (plate with a hole) immersed in a fluidof intermediate
permittivity will experience a meta-stable equilibrium (restoring force) near
the center of the hole. Stability is the result of a geometry-induced
transition in the sign of the force, from repulsive to attractive, that occurs
as the disk approaches the hole---in planar or nearly-planar geometries, the
same material combination yields a repulsive force at all separations in
accordance with the Dzyaloshinskii--Lifshitz--Pitaevskii condition of
fluid-induced repulsion between planar bodies. We explore the stability of the
system with respect to rotations and lateral translations of the disks, and
demonstrate interesting transitions (bifurcations) in the rotational stability
of the disks as a function of their size. Finally, we consider the reciprocal
situation in which the disk--plate materials are interchanged, and find that in
this case the system also exhibits meta-stability. The forces in the system are
sufficiently large to be observed in experiments and should enable measurements
based on the diffusion dynamics of the suspended bodies.
