  In this paper we demonstrate that solitons of a simple real scalar field
model that are {\it static and linearly stable} do exist when considered in a
(3+1)-dimensional, spatially compact space-time background, the static Einstein
universe, which is a good approximation to the observed universe for
sufficiently small time intervals. We study the properties of these solutions
for a $\Phi^4$-potential and demonstrate that next to the fundamental
solutions, excited configurations exist. We also investigate general
perturbations about the solitons, determine their eigenfrequency spectra and
compare them with those of the perturbations about the vacua of the model. We
find that the degeneracy with respect to the multipoles of the perturbation,
which is present for the vacua, does no longer exist in the presence of the
soliton. Moreover, specific perturbations correspond to zero modes of the
system. Our results have applications in condensed matter physics as well as
computations of quantum effects (e.g. the Casimir energy) in spatially compact
space-times in the presence of soliton-like objects.
