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

Abstract

In this paper, our aim is to study the geometry of [Formula: see text]-dimensional trapped and marginally trapped submanifolds immersed in a Lorentzian space form [Formula: see text] of constant sectional curvature [Formula: see text]. In this setting, we establish sufficient conditions to guarantee that a complete trapped submanifold with parallel mean curvature vector in [Formula: see text] must be pseudo-umbilical. 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 [Formula: see text]-dimensional complete pseudo-umbilical marginally trapped submanifold of [Formula: see text] with parallel mean curvature vector is, in fact, totally umbilical.