Fábio R. dos Santos

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 L1n+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 L1n+p(c) 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 n-dimensional complete pseudo-umbilical marginally trapped submanifold of L1n+p(c) with parallel mean curvature vector is, in fact, totally umbilical.

On the first stability eigenvalue of hypersurfaces in the Euclidean and hyperbolic spaces

Abstract In this paper, we obtain upper bounds for the first eigenvalue of the stability operator of a closed constant mean curvature hypersurface ∑n immersed either in the Euclidean space ℝn+1 or in the hyperbolic space ℍn+1, n ≥ 2, in terms of the mean curvature and the length of the total umbilicity operator of ∑n. As application, we derive a nonexistence result concerning strong stable hypersurfaces in these ambient spaces. Furthermore, through the calculus of the first stability eigenvalue of circular cylinders in ℝn+1 and of hyperbolic cylinders in ℍn+1, we present a conjecture related to the first stability eigenvalue of complete constant mean curvature hypersurfaces immersed either in ℝn+1 or in ℍn+1.