  In a recent paper, Eichmair, Galloway and Pollack have proved a
Gannon-Lee-type singularity theorem based on the existence of marginally outer
trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic
spacetimes. However, one might wonder whether the corresponding incomplete
geodesics could still be complete in a possible non-globally hyperbolic
extension of spacetime. In this note, some variants of that result are given
with weaker causality assumptions, thus suggesting that the answer is
generically negative, at least if the putative extension has no closed timelike
curves. However, unlike in the case of MOTS, on which only the outgoing family
of normal geodesics is constrained, we have found it necessary in our proofs to
impose also a weak convergence condition on the ingoing family of normal
geodesics. In other words, we consider marginally trapped surfaces (MTS) in
chronological spacetimes, introducing the natural notion of a generic MTS. In
particular, a Hawking-Penrose-type singularity theorem is proven in
chronological spacetimes with dimensions greater than 2 containing a generic
MTS. Such surfaces naturally arise as cross-sections of quasi-local
generalizations of black hole horizons, such as dynamical and trapping
horizons. We end with some comments on the existence of MTS in initial data
sets.
