On the geometry of trapped and marginally trapped submanifolds in Lorentzian space forms

In this paper, our aim is to study the geometry of n-dimensional trapped and marginally trapped submanifolds immersed in a Lorentzian space form L-1(n+p) (c) of constant sectional curvature c. In this setting, we establish sufficient conditions to guarantee that a complete trapped submanifold with parallel mean curvature vector in L-1(n+p) (c) must be pseudoumbilical. Afterwards, we obtain a nonexistence result concerning complete trapped submanifolds in the Lorentz-Minkowski space. Furthermore, under suitable constraints on the Ricci curvature and the second fundamental form, we show that an n-dimensional complete pseudo-umbilical marginally trapped submanifold of L-1(n+p) (c) with parallel mean curvature vector is, in fact, totally umbilical.