- The role played by the sophisticated scalar potential, dictated by spontaneous compactification, is analyzed. A fine-tuning is mandatory for achieving asymptotic flatness. Two main aspects are studied. (i) The three-fold spherically symmetric case exhibits localized four-dimensional objects, to be referred to as ``gravitational bags''. These are cores of scalar fields confined by means of a cosmic domain wall, whose size only slightly exceeds equal-mass black holes. (ii) The cosmological case introduces a novel scenario of ``solitary evolution''. Triggered by the collapse of the extra dimensions, the universe undergoes inflationary stage before setting in an oscillating fashion, in its ground state.