Minimal surfaces, spatial topology and singularities in space-time

Using an approach first advocated by Gannon (1975, 1976) and recent results of Meeks, Simon and Yau (1982) on the existence of compact minimal surfaces some new results are obtained relating non-trivial spatial topology to the occurrence of singularities in space-time. For example, it is shown that if V³ is a contracting body with mean-convex boundary homeomorphic to a two-sphere in a space-time M⁴ obeying appropriate curvature and causality assumptions, then either V³ is a three-cell or M ⁴ is non-space-like geodesically incomplete.