When does curved spacetime guarantee a singularity must form?

Penrose's singularity theorem uses trapped surfaces — regions where even outgoing light rays converge — to prove singularities must form. This work tests whether that logic extends to higher-dimensional analogues in spacetimes of arbitrary dimension. The curvature conditions needed turn out to be geometry-dependent rather than universal, meaning some higher-dimensional trapped surfaces can still signal singularities, but they might lie inside observable regions rather than safely behind horizons.