Geometric analysis of the Lorentzian distance function on trapped submanifolds

We use geometric analysis of the Lorentzian distance on an n-dimensional spacetime to derive some non-existence results and some sharp mean curvature estimates for trapped submanifolds contained in the chronological future of either a point or of a space-like achronal hypersurface . Our results are given as an application of the corresponding Hessian comparison results for the Lorentzian distance function and the so-called weak maximum principle for the Laplacian operator.