We study the motion of an n-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power p of the mean curvature. We prove that for any p ∈ (0, 1], the flow exists for all time when the Ricci tensor of the ambient space is bounded from below on the set of timelike unit vectors. Moreover, if we ass...

We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time.

We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time.

Given a globally hyperbolic spacetime M , we show the existence of a smooth spacelike Cauchy hypersurface S and, thus, a global diffeomorphism between M and R × S.

For an (m+1)-dimensional space-time (X, g), define a mapped null hypersurface to be a smooth map ν : N → X (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and g-orthogonal to ν. We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle ST M of an immersed spacelike hypersurf...

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...

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained...

Taking the triangle areas as independent variables in the theory of Regge calculus can lead to ambiguities in the edge lengths, which can be interpreted as discontinuities in the metric. We construct solutions to area Regge calculus using a triangulated lattice and find that on a spacelike hypersurface no such discontinuity can arise. On a null hypersurface however, we can have such a situation...

We present a detailed examination of the variational principle for metric general relativity as applied to a " quasilocal " spacetime region M (that is, a region that is both spatially and temporally bounded). Our analysis relies on the Hamiltonian formulation of general relativity, and thereby assumes a foliation of M into spacelike hypersurfaces Σ. We allow for near complete generality in the...

Abstract We give conditions for a conformal vector field to be tangent null hypersurface. particularize two important cases: Killing and closed field. In the first case, we obtain result ensuring that hypersurface is horizon. second one, gives rise foliation of manifold by totally umbilical hypersurfaces with constant mean curvature which can spacelike, timelike or null. prove several results e...