Complete Spacelike Hypersurfaces Research Articles

Some maximum principles for the drift Laplacian applied to complete spacelike hypersurfaces

Some maximum principles for the drift Laplacian applied to complete spacelike hypersurfaces

Uniqueness and nullity of complete spacelike hypersurfaces immersed in the anti-de Sitter space

Uniqueness and nullity of complete spacelike hypersurfaces immersed in the anti-de Sitter space

Complete spacelike hypersurfaces in the anti-de Sitter space: rigidity, nonexistence and curvature estimates

Our purpose is to investigate the geometry of complete spacelike hypersurfaces immersed in the anti-de Sitter space $\mathbb H_1^{n+1}$. We start by proving rigidity results for such hypersurfaces under suitable constraints on their higher order mean curv

Complete spacelike hypersurfaces with constant scalar curvature: Descriptions and gaps

We provide sharp lower and upper bounds for the supremum of the norm of the total umbilicity tensor of complete spacelike hypersurfaces with constant scalar curvature immersed in a Lorentzian space form and satisfying a suitable Okumura-type inequality, which corresponds to a weaker hypothesis when compared with the geometric condition of the hypersurface having two distinct principal curvatures. Furthermore, we give a complete description and the gaps of the spacelike hypersurfaces which realize our estimates, obtaining as a consequence new characterizations of totally umbilical spacelike hypersurfaces and hyperbolic cylinders of Lorentzian space forms. Our approach is based on a version of Omori–Yau’s maximum principle for trace-type differential operators defined on a complete Riemannian manifold.

Uniqueness and nonexistence of complete spacelike hypersurfaces, Calabi–Bernstein type results and applications to Einstein–de Sitter and steady state type spacetimes

We investigate the geometry of complete spacelike hypersurfaces (immersed) in a generalized Robertson-Walker spacetime $$-I\times _fM^{n}$$ . Under suitable constraints on the sectional curvature of the Riemannian fiber $$M^n$$ , on the warping function f and on the future mean curvature (that is, the mean curvature function with respect to the future-pointing Gauss map of the spacelike hypersurface), we are able to prove that such a spacelike hypersurface must be a slice $$\{t\}\times M^{n}$$ of the ambient spacetime. Nonexistence and Calabi–Bernstein type results concerning entire spacelike graphs constructed over the Riemannian fiber $$M^n$$ are also obtained, as well as applications to the Einstein–de Sitter and steady state type spacetimes are given. Our approach is based on the so-called Omori–Yau’s generalized maximum principle and on certain integrability properties due to Yau.

Constant mean curvature spacelike hypersurfaces in standard static spaces: rigidity and parabolicity

Our purpose in this paper is investigate the geometry of complete constant mean curvature spacelike hypersurfaces immersed in a standard static space, that is, a Lorentzian manifold endowed with a globally defined timelike Killing vector field. In this setting, supposing that the ambient space is a warped product of the type $M^n\times_{\rho}\mathbb{R}_1$ whose Riemannian base $M^n$ has nonnegative sectional curvature and the warping function $\rho$ is convex on $M^n$, we use the generalized maximum principle of Omori-Yau in order to establish rigidity results concerning these spacelike hypersurfaces. We also study the parabolicity of maximal spacelike surfaces in $M^2\times_{\rho}\mathbb{R}_1$ and we obtain uniqueness results for entire Killing graphs constructed over $M^n$.

Complete spacelike hypersurfaces immersed in pp-wave spacetimes

We study some aspects of the geometry of complete spacelike hypersurfaces immersed into a pp-wave spacetime, namely, into a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying suitable versions of the classical Hopf and Stokes theorems and a criterion of parabolicity for complete Riemannian manifolds, we obtain sufficient conditions which guarantee that a complete spacelike hypersurface is either maximal, 1-maximal or totally geodesic. As a consequence of these results, we also establish some results of nonexistence concerning such spacelike hypersurfaces. Finally, considering constant mean curvature closed spacelike hypersurfaces immersed in a pp-wave spacetime, we study a notion of stability via the first nonzero eigenvalue of the Laplacian.

New characterizations of spacelike hyperplanes in the steady state space

In this article, we deal with complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb {S}^{n+1}_{1}$ which is known as the steady state space $\mathcal {H}^{n+1}$. Under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we are able to prove that they must be spacelike hyperplanes of $\mathcal {H}^{n+1}$. Furthermore, through the analysis of the hyperbolic cylinders of $\mathcal {H}^{n+1}$, we discuss the importance of the main hypothesis in our results. Our approach is based on a generalized maximum principle at infinity for complete Riemannian manifolds.

New Calabi–Bernstein type results in Lorentzian product spaces with density

We investigate the rigidity of complete spacelike hypersurfaces immersed in a weighted Lorentzian product space of the type R1×Pfn, endowed with a weight function f which does not depend on the parameter t∈R and whose Riemannian fiber Pn has nonnegative Bakry–Émery–Ricci tensor. In this direction, supposing that the f-mean curvature is constant and assuming appropriate constraints on the norm of the gradient of f, we prove that such a spacelike hypersurface must be a slice of the ambient space. As application of our investigation, we obtain new Calabi–Bernstein type results concerning entire spacelike graphs constructed over Pn. Our approach is based on an extension for the drift Laplacian of a generalized maximum principle of Akutagawa (1987) and a Liouville-type result due to Pigola et al. (2005).

On Geometric Analysis of the Dynamics of Volumetric Expansion and its Applications to General Relativity

In this paper, we discuss the global aspect of the geometric dynamics of volumetric expansion and its applications to the problem of the existence in the space-time of compact and complete spacelike hypersurfaces and to the global geometry of generalized Robertson–Walker space-times.

Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space

Abstract Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space, Pacific J. Math. 292 (2018), 1–19].

Calabi–Bernstein type results for complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes

In this article we study complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes. Under appropriate geometric assumptions we prove new parametric uniqueness results as well as obtain very general Calabi–Bernstein type results for the constant mean curvature equation for spacelike graphs in these ambient spacetimes.

Spacelike Hypersurfaces in Spatially Parabolic Standard Static Spacetimes and Calabi–Bernstein-Type Problems

Complete spacelike hypersurfaces in spatially parabolic standard static spacetimes are studied. Under natural boundedness assumptions, we show how the parabolicity of the base is inherited by any spacelike hypersurface and vice versa. Moreover, we give new uniqueness and non-existence results for complete spacelike hypersurfaces in these ambient spacetimes as well as solve new Calabi–Bernstein-type problems.

From holography to the geometry of the spacetime

We show the geometric consequences that the holographic principle has on the spacetime. Namely, we prove that complete spacelike hypersurfaces in a spacetime that satisfies the holographic principle are non-parabolic. This has important consequences on the behaviour of the Brownian motion in these spacelike hypersurfaces, as well as provides a method for finding examples of universes that lead to a violation of this principle.

Constant Mean Curvature Spacelike Hypersurfaces in Spatially Open GRW Spacetimes

In this paper we provide under certain geometric and physical assumptions new uniqueness and non-existence results for complete spacelike hypersurfaces of constant mean curvature in spatially open Generalized Robertson-Walker spacetimes. Some of our results are then applied to relevant spacetimes as the steady state spacetime, Einstein-de Sitter spacetime and certain radiation models.

$$\phi $$ ϕ -Parabolicity and the Uniqueness of Spacelike Hypersurfaces Immersed in a Spatially Weighted GRW Spacetime

In this paper, we extend a technique due to Romero et al. (Class Quantum Gravity 30:1–13, 2013; Int J Geom Methods Mod Phys 10:1360014, 2013; J Math Anal Appl 419:355–372, 2014) establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime whose fiber has $$\phi $$ -parabolic universal Riemannian covering. As some applications of this criteria, we obtain uniqueness results concerning spacelike hypersurfaces immersed in spatially weighted generalized Robertson–Walker spacetimes. Furthermore, Calabi–Bernstein type results are also given.

A New Calabi–Bernstein Type Result in Spatially Closed Generalized Robertson–Walker Spacetimes

The aim of this article is to study the uniqueness of a complete spacelike hypersurface $${\sum^{n}}$$ immersed with constant mean curvature H in a spatially closed generalized Robertson–Walker spacetime $${\overline{M}^{n+1} = -I {\times_{f}} {M^{n}}}$$ , whose Riemannian fiber $${M^n}$$ has positive curvature. Supposing that the warping function f is such that −log f is convex and $${H{f^{\prime}} \leq 0}$$ along $${\sum^{n}}$$ , we show that $${\sum^{n}}$$ must be isometric to a totally geodesic slice of $${\overline{M}^{n+1}}$$ . When $${\overline{M}^{n+1}}$$ is a Lorentzian product space, we obtain a new Calabi–Bernstein type result concerning the CMC spacelike hypersurface equation.

On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space

In this paper, we study complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space. By applying some generalized maximum principles to a suitable Cheng–Yau modified operator, we establish sufficient conditions which assure that such a spacelike hypersurface must be a totally umbilical hypersurface of the ambient space.

Complete spacelike hypersurfaces in orthogonally splitted spacetimes

We provide some “half-space theorems” for spacelike complete non-compact hypersurfaces into orthogonally splitted spacetimes. In particular we generalize some recent work of Rubio and Salamanca on maximal spacelike compact hypersurfaces. Beside compactness, we also relax some of their curvature assumptions and even consider the case of nonconstant mean curvature bounded from above. The analytic tools used in various arguments are based on some forms of the weak maximum principle.

Bernstein type results for complete spacelike hypersurfaces immersed in a weighted conformally stationary spacetime

In this paper, we deal with complete noncompact spacelike hypersurfaces immersed in a weighted conformally stationary spacetime endowed with a closed conformal timelike vector field [Formula: see text]. Under suitable constraints on the weighted mean curvature of such a spacelike hypersurface, we establish sufficient conditions to ensure that it must be an integral leaf of the foliation orthogonal to [Formula: see text].