Abstract: Inspired by the classical singularity theorems in General Relativity, Galloway and Ling (Commun. Math. Phys. 2017, 2024) have shown the following: If a globally hyperbolic spacetime satisfying the null energy condition contains a closed, spacelike Cauchy surface N which is  strictly 2-convex (meaning that the sum of the lowest two eigenvalues of the future extrinsic curvature is positive), then N is either a spherical space or past null geodesically incomplete.

In recent work (with Eric Ling, Carl Rossdeutscher and Roland Steinbauer) we have relaxed the above convexity condition in essentially two respects. Firstly, we admit 2-convex extrinsic curvatures (for which the sum of the lowest two eigenvalues is non-negative). Secondly, if N admits a U(1) isometry group, we can impose even less restricive convexity conditions. In addition to spherical spaces and  past null geodesically incompleteness, this allows for certain classes of surface bundles for N or a finite cover therof, which we classify.  While our results do not use Einstein's equations, we provide several classes of vacuum solutions (with our without cosmological constant) of these equations as examples.

In my talk I will give a brief overview of singularity theorems and then focus on explaining the above results via examples. The proofs will only be outlined.