Minimal surfaces, spatial topology and singularities in space-time

Abstract

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 M4 is non-space-like geodesically incomplete.