Publications

We deal with n-dimensional spacelike submanifolds immersed with parallel mean curvature vector h in a pseudo-Riemannian space form  𝐿_𝑞^{𝑛+𝑝}(𝑐) of index 1 ≤ q ≤ p and constant sectional curvature c ∈{− 1,0,1}. Considering the cases when h is either spacelike or timelike, we are able to prove that such a spacelike submanifold is either totally umbilical or it holds a lower estimate for the supremum of the norm of its traceless second fundamental form, occurring equality if the spacelike submanifold is pseudo-umbilical and its principal curvatures are constant. In our approach, we use three main core concepts: Stochastic completeness, parabolicity and L1-Liouville property. 

Our purpose is to establish new gap type and characterization results concerning n-dimensional spacelike submanifolds immersed with parallel mean curvature vector in the (𝑛+𝑝)-dimensional de Sitter space 

𝑆_𝑞^{𝑛+𝑝} of index q (1≤𝑞≤𝑝). Initially, by applying a weak form of the Omori–Yau maximum principle, we obtain sufficient conditions which guarantee that a stochastically complete spacelike submanifold 

𝑀^𝑛 immersed in 𝑆_𝑞^{𝑛+𝑝} is either totally umbilical or isometric to a maximal isoparametric spacelike submanifold. Furthermore, by assuming that either the Hilbert–Schmidt norm of the traceless second fundamental form of 𝑀^𝑛 converges to zero at infinity or that 𝑀^𝑛 has polynomial volume growth, we provide a set of geometric hypotheses which guarantee the umbilicity of 𝑀^𝑛.

Under the hypothesis that the second fundamental form has finite 𝐶1 norm, we show that the n-dimensional spacelike hyperplanes of the (𝑛+𝑝)-dimensional pseudo-Euclidean space 𝑅_𝑝^𝑛+𝑝 of index p are the only complete spacelike submanifolds 𝑋:𝑀𝑛↬𝑅_𝑝^𝑛+𝑝 having polynomial f-volume growth, for 𝑓=|𝑋|^2/4, and parallel Gaussian mean curvature vector 𝜉, with|𝜉|^2 ≤ 2𝑝. When 𝑋:𝑀^𝑛 ↬ 𝑅_𝑝^𝑛+𝑝 is complete noncompact, supposing that the norm of the second fundamental form converges to zero at infinity, we also conclude that it must be an n-dimensional spacelike hyperplane of 𝑅_𝑝^𝑛+𝑝.

Our aim in this paper is to study the uniqueness of complete spacelike hypersurfaces immersed in the anti-de Sitter space 

𝐻_1^{𝑛+1}, through the behavior of their higher order mean curvatures. This is done by applying a suitable maximum principle concerning smooth vector fields whose norm is Lebesgue integrable on a complete Riemannian manifold. We also infer the nullity of complete r-maximal spacelike hypersurfaces and, in particular, we establish a nonexistence result concerning complete 1-maximal spacelike hypersurfaces in 𝐻_1^{𝑛+1}. 

In this paper, we deal with $n$-dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space $L_{p}^{n+p}$ of index $p>1$, which obeys some curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). Under appropriate hypothesis, we are able to prove that such a spacelike submanifold is either totally umbilical or isometric to an isoparametric submanifold of the ambient space. For this, we use three main core analytical tools: a suitable version of the Omori--Yau maximum principle, parabolicity with respect to a modified Cheng--Yau operator and a certain integrability property. 

We investigate the umbilicity of n-dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field in the de Sitter space Sn p+p of index p > 1. We recall that a spacelike submanifold is said to be linear Weingarten when its mean curvature function H and its normalized scalar curvature R satisfy a linear relation of the type R = aH +b, for some constants a, b ∈ R. Under suitable constraints on the values of a and b, we apply a generalized maximum principle for a modified Cheng–Yau operator L in order to show that such a spacelike submanifold must be either totally umbilical or isometric to a product M1×M2×· · ·×Mk, where the factors Mi are totally umbilical submanifolds of Sn p+p which are mutually perpendicular along their intersections. Moreover, we also study the case in which these spacelike submanifolds are L-parabolic.

Our purpose is to investigate the geometry of complete spacelike hypersurfaces immersed in the anti-de Sitter space H^n+1_1. We start by proving rigidity results for such hypersurfaces under suitable constraints on their higher order mean curvatures. We also obtain a lower estimate for the index of minimum relative nullity for r-maximal spacelike hypersurfaces and a nonexistence result for 1-maximal spacelike hypersurfaces of H^n+1_1. Finally, we employ a technique due to Aledo and Alías (2000) to prove some curvature estimates for complete spacelike hypersurface of H^n+1_1; as a consequence, we get further nonexistence results. In particular, we show the nonexistence of complete maximal spacelike hypersurfaces in certain open regions of H^n+1_1. Our approach is mainly based on a suitable extension of the generalized maximum principle of Omori and Yau due to Alías, Impera and Rigoli (2012). 

We deal with 𝑛-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form 𝐿_𝑞^{𝑛+𝑝}(𝑐) of index 1≤𝑞≤𝑝 and constant sectional curvature 

𝑐∈{−1,0,1}. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento.

Let Mn be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean  curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space L_p^{n+p} of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold Mn be either totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold L_1^{n+1} → L_p^{n+p}, with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori–Yau’s generalized maximum principle for a Cheng–Yau’s modified operator.