On complete trapped submanifolds in globally hyperbolic spacetimes

Abstract

The aim of this manuscript is to obtain rigidity and non-existence results for complete (non necessarily compact) spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes. We also obtain results regarding the geometry of such submanifolds by ensuring, under some mild hypothesis, the non-existence of local minima or maxima of certain distinguished function.
As an application in General Relativity, we obtain several nice results regarding (nonnecessarily closed) trapped surfaces in a huge family of spacetimes. In fact, we show how our technique allows us to recover in particular some relevant previous results for trapped surfaces in both, standard static spacetimes and Generalized Robertson-Walker spacetimes.