Please note special time.  We study time-like constant mean curvature hypersurfaces of Minkowski space from the point of view of their initial value problems. We focus on initial data close to those generating spherically symmetric expanding solutions, such as anti de Sitter space. Using intuitions from mathematical relativity, we explain how a ``global stability'' statement cannot be true due to the presence of cosmological horizons. Yet using the same intuition we discuss how to easily obtain a spatially localised stability result after a careful geometric reformulation of the problem.