Rigidity aspects of a cosmological singularity theorem

Improving a singularity theorem in General Relativity by Galloway and Ling we show the following (cf.\ Theorem 1): If a globally hyperbolic spacetime $M$ satisfying the null energy condition contains a closed, spacelike Cauchy surface $(V,g,K)$ (with metric $g$ and extrinsic curvature $K$) which is 2-convex (meaning that the sum of the lowest two eigenvalues of $K$ is non-negative), then either $M$ is past null geodesically incomplete, or $V$ is a spherical space, or $V$ or some finite cover is a surface bundle over the circle, with totally geodesic fibers. Moreover, (cf.\ Theorem 2) if $(V,g,K)$ admits a $U(1)$ isometry group with corresponding Killing vector $\xi$, we can relax the convexity requirement in terms of a decomposition of $K$ with respect to the directions parallel and orthogonal to $\xi$. Finally, (cf. Propositions 1-3) in the special cases that $V$ is either non-orientable, or non-prime, or an orientable Haken manifold with vanishing second homology, we obtain stronger statements in both Theorems without passing to covers.