Carl Rossdeutscher: Rigidity aspects of a singularity Theorem

In 2018 Galloway and Ling established the following cosmological singularity theorem: If a (3+1)-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface with a positive definite second fundamental form (i.e., it’s expanding in all directions), then the spacetime is past null geodesically incomplete unless the Cauchy surface is a spherical space. We present some rigidity results for this singularity theorem. In particular if the second fundamental form is only positive semidefinite and the spacetime is past null geodetically complete, we show that the Cauchy surface (or at least a finite cover thereof) is a surface bundle over the circle with totally geodesic fibers or a spherical space. Under certain additional assumptions on the Cauchy surface, we show that passing to a cover is unnecessary. Our results make in particular use of the positive resolution of the virtual positive first Betti number conjecture by Agol. If a spacetime admits a U(1) isometry group, we can relax the assumption on the second fundamental form further.