Rigid singularity theorem in globally hyperbolic spacetimes

We show the rigid singularity theorem, that is, a globally hyperbolic spacetime satisfying the strong energy condition and containing future or past trapped sets, either is timelike geodesically incomplete or splits isometrically as $\mathrm{space}\times \mathrm{time}.\mathrm{}$This result is related to Yau’s Lorentzian splitting conjecture.