Stability of Cauchy Horizons

We prove that for any 3-dimensional compact hypersurface S in a noncompact 4-dimensional space-time manifold M, S ⊂ M, the set of Lorentzian metrics on M for which S is a partial Cauchy surface and Cauchy horizon of S is nonempty contains a nonempty open subset (in compact-open topology). This result indicates that the set of metrics admitting Cauchy horizons originating from compact hypersurfaces is large.