biharmonic surfaces in euclidean space $mathbb{e}^3$ are firstly studied from a differential geometric point of view by bang-yen chen, who showed that the only biharmonic surfaces are minimal ones. a surface $x : m^2rightarrowmathbb{e}^{3}$ is called biharmonic if $delta^2x=0$, where $delta$ is the laplace operator of $m^2$. we study the $l_k$-biharmonic spacelike hypersurfaces in the $4$-dimen...

Given a globally hyperbolic spacetime, the existence of a (smooth) spacelike Cauchy hypersurface S has been proven recently. Here, we prove that any acausal spacelike compact submanifold with boundary can be smoothly extended to a spacelike Cauchy hypersurface. Apart from the interest as a purely geometric question (applicable to the Cauchy problem in General Relativity), the result is motivate...

A criterion for a spacelike hypersurface in a conformally stationary spacetime to be a Cauchy hypersurface is given. The criterion is based on the forward and the backward completeness of a non-reversible Finsler metric associated to the hypersurface.

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M , have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, b...

we study connected orientable spacelike hypersurfaces $x:m^{n}rightarrowm_q^{n+1}(c)$, isometrically immersed into the riemannian or lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~l_kx=ax+b$,~ where $l_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $h_{k+1}$ of the hypersurface for a fixed integer $0leq k

We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

In this paper we consider a variational problem for spacelike hypersurfaces in the (n + 1)-dimensional Lorentz-Minkowski space L, whose critical points are hypersurfaces supported in a spacelike hyperplane determined by two facts: the mean curvature is a linear function of the distance to and the hypersurface makes a constant angle with along its boundary. We prove that the hypersurface is rota...

Given any (topological) Cauchy hypersurface S of a globally hyperbolic spacetime, we construct a smooth function τ : M → R such that the levels St = τ (t), t ∈ R satisfy: (i) S = S0, (ii) each t ∈ R\{0} is a (smooth) spacelike Cauchy hypersurface. If S is also acausal (and not only achronal) then the smooth function τ becomes a time function, i.e., it is strictly increasing on any future-direct...

Recommended by Christian Corda We obtain a height estimate concerning to a compact spacelike hypersurface Σ n immersed with constant mean curvature H in the anti-de Sitter space H n1 1 , when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space H n. Our estimate depends only on the value of H and on the geometry of ∂Σ...

In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σn immersed in the (n + 1)dimensional Lorentz–Minkowski space Ln+1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of Ln+1. Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of Ln+1. © 2007 Els...