  We investigate the behavior of identical dipolar fermions with aligned dipole
moments in two-dimensional multilayers at zero temperature. We consider density
instabilities that are driven by the attractive part of the dipolar interaction
and, for the case of bilayers, we elucidate the properties of the stripe phase
recently predicted to exist in this interaction regime. When the number of
layers is increased, we find that this "attractive" stripe phase exists for an
increasingly larger range of dipole angles, and if the interlayer distance is
sufficiently small, the stripe phase eventually spans the full range of angles,
including the situation where the dipole moments are aligned perpendicular to
the planes. In the limit of an infinite number of layers, we derive an analytic
expression for the interlayer effects in the density-density response function
and, using this result, we find that the stripe phase is replaced by a collapse
of the dipolar system.
