Spacelike Hypersurfaces Of Constant Mean Curvature Research Articles

On Calabi–Bernstein’s Problem for Uniqueness of Complete Spacelike Hypersurfaces in Weighted Generalized Robertson–Walker Spacetimes

In this paper, we study Calabi–Bernstein’s problem for the uniqueness of complete constant weighted mean curvature spacelike hypersurfaces Σn in weighted generalized Robertson–Walker spacetimes −I×ρMfn. Under appropriate geometric assumptions, we confirm that Σn is a spacelike slice.

The Intrinsic Structure of High-Dimensional Data According to the Uniqueness of Constant Mean Curvature Hypersurfaces

In this paper, we study the intrinsic structures of high-dimensional data sets for analyzing their geometrical properties, where the core message of the high-dimensional data is hiding on some nonlinear manifolds. Using the manifold learning technique with a particular focus on the mean curvature, we develop new methods to investigate the uniqueness of constant mean curvature spacelike hypersurfaces in the Lorentzian warped product manifolds. Furthermore, we extend the uniqueness of stochastically complete hypersurfaces using the weak maximum principle. For the more general cases, we propose some non-existence results and a priori estimates for the constant higher-order mean curvature spacelike hypersurface.

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 $M^n$.

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.

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.

Characterizations of complete CMC spacelike hypersurfaces satisfying an Okumura type inequality

We deal with complete constant mean curvature spacelike hypersurfaces immersed in a Lorentzian space form and satisfying a suitable Okumura type inequality, which is a weaker hypothesis than to assume that the spacelike hypersurface has two distinct principal curvatures. In this setting, we obtain estimates for the norm of the traceless part of the second fundamental form and we show that these estimates are sharp in the sense that the hyperbolic cylinders realize them.

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\).

On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field

We study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces whose constant mean curvature is non-zero and also that every compact maximal hypersurface is totally geodesic. Moreover, we give an extension of the classical Calabi–Bernstein theorem to this class of pp-wave spacetimes.

Complete spacelike hypersurfaces in generalized Robertson–Walker and the null convergence condition: Calabi–Bernstein problems

We study constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker spacetimes $$\overline{M}= I \times _f F$$ which are spatially parabolic (i.e. its fiber F is a (non-compact) complete Riemannian parabolic manifold) and satisfy the null convergence condition. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi–Bernstein type results are given.

ON THE STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER ORDER MEAN CURVATURE IN A DE SITTER SPACE

Abstract. The closed spacelike hypersurfaces with higher order meancurvature is discussed in a de Sitter space. The hypersurface is provedstable if and only if it is totally umbilical. 1. IntroductionAs we all know, the hypersurfaces with constant mean curvature or con-stant scalar curvature in real space forms are characterized as critical points ofthe area functional for volume-preserving variations. Many results have beenachieved about hypersurfaces with constant mean curvature or constant scalarcurvature in a unit sphere S n+1 (1) [1, 2, 3]. Among these results, the geodesicsphere is the only stable compact hypersurface with constant mean curvaturein a sphere as in [3]. After that, the closed hypersurfaces with higher ordermean curvature immersed in a Riemannian space form are studied and similarresults are obtained by other researches [4, 8, 12].Achievements are not only obtained in Riemannian space, in fact, many re-searches are also conducted in Lorentzian spaces. Constant mean curvaturespacelike hypersurfaces are solutions to a variational problems. Actually, theyare the critical points of the area functional for variations that leave constanta certain volume function. In this sense, Barbosa and Oliker [5] computed thesecond variation formula and obtained in the de Sitter space S

Complete Constant Mean Curvature Spacelike Hypersurfaces in the Einstein-De Sitter Spacetime

The uniqueness and nonexistence results on complete constant mean curvature spacelike hypersurfaces lying between two spacelike slices in the Einstein–de Sitter spacetime are given. They are obtained from a Liouville-type theorem applied to a distinguished smooth function on a complete constant mean curvature hypersurface.

Constant mean curvature spacelike hypersurfaces in Lorentzian warped products and Calabi–Bernstein type problems

In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a large family of uniformly elliptic EDP’s, so solving the Calabi–Bernstein problem in a wide family of spacetimes.

On the umbilicity of complete constant mean curvature spacelike hypersurfaces

In this paper, we show that a complete spacelike hypersurface immersed with constant mean curvature either in the de Sitter space or in the anti-de Sitter space must be totally umbilical, provided that its Gauss mapping has some suitable behavior. In particular, we use an extension of Hopf’s maximum principle due to Yau in order to give new positive answers to Goddard’s conjecture.

Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field

A new technique to study spacelike hypersurfaces of constant mean curvature in a spacetime which admits a timelike gradient conformal vector field is introduced. As an application, the leaves of the natural spacelike foliation of such spacetimes are characterized in some relevant cases. The global structure of this class of spacetimes is analyzed and the relation with its well-known subfamily of generalized Robertson–Walker spacetimes is exposed in detail. Moreover, some known uniqueness results for compact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes are widely extended. Finally, and as a consequence, several Calabi–Bernstein problems are solved obtaining all the entire solutions on a compact Riemannian manifold to the constant mean curvature spacelike hypersurface equation, under natural geometric assumptions.

Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces

We use the half-space model for the open set of a de Sitter space associated to the steady state space to obtain some sharp a priori estimates for the height and the slope of certain constant mean curvature spacelike graphs. These estimates allow us to prove some existence and uniqueness theorems about complete non-compact constant mean curvature spacelike hypersurfaces in de Sitter spaces with prescribed asymptotic future boundary. Their geometric properties are studied.

Spacelike hypersurfaces of constant mean curvature with free boundary in Lorentzian space forms

In this paper we introduce a variational problem for spacelike hypersurfaces in Lorentzian space forms whose critical points turn out to be constant mean curvature spacelike hypersurfaces which intersect a given support hypersurface at a constant hyperbolic angle. For the two-dimensional case, when the support surface is totally umbilical we prove that only such surfaces are totally umbilical too.

On constant mean curvature spacelike hypersurfaces in Lorentz manifolds

On constant mean curvature spacelike hypersurfaces in Lorentz manifolds

Constant mean curvature spacelike hypersurfaces with spherical boundary in the Lorentz-Minkowski space

We study compact spacelike hypersurfaces (necessarily with non-empty boundary) with constant mean curvature in the ( n + 1)-dimensional Lorentz-Minkowski space. In particular, when the boundary is a round sphere we prove that the only such hypersurfaces are the hyperplanar round balls (with zero mean curvature) and the hyperbolic caps (with non-zero constant mean curvature).

Existence of CMC Cauchy surfaces and spacetime splitting

In this paper, we review results on the existence (and nonexistence) of constant mean curvature spacelike hypersurfaces in the cosmological setting, and discuss the connection to the spacetime splittng problem. It is a pleasure to dedicate this paper to Robert Bartnik, who has made fundamental contributions to this area.