Classification of f-biharmonic submanifolds in Lorentz space forms

In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

Geometric Study of Marginally Trapped Surfaces in Space Forms and Robertson-Walker Spacetimes—An Overview

A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classifications of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector field, having finite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions.

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

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.