The paper estimates curvature for specific hypersurfaces in a special space.
problem Estimating curvature for spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Analyzing the geometry of spacelike admissible graphic hypersurfaces.
result Existence of specific hypersurfaces with prescribed curvature and boundary conditions.
The paper studies how certain spacelike surfaces evolve over time in a specific space.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving surfaces converge to a hyperbolic plane as time goes to infinity.
The paper studies a flow of spacelike surfaces in Lorentz-Minkowski space, proving convergence to a hyperbolic plane.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Anisotropic inverse mean curvature flow with Neumann boundary condition.
result The flow converges to a hyperbolic plane as time tends to infinity.
The paper studies how spacelike surfaces evolve in Lorentz-Minkowski space over time.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Inverse Gauss curvature flow with Neumann boundary condition.
result The evolving surfaces converge to a constant function as time goes to infinity.
The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
problem Conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
method Proving Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in Euclidean and Minkowski contexts.
result Zero Gaussian curvature convex hypersurfaces must be hyperplanes if the mean curvature goes to zero at infinity.
The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.
problem Analyzing Bernstein property for minimal spacelike surfaces in 4D Minkowski space.
method Study of an extension of the Bernstein Theorem for minimal spacelike surfaces in R^4_1.
result The Bernstein property does not hold in general for graphic spacelike surfaces in R^4_1.
Similar to the definition of Dupin hypersurface in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of conformal geometry. Further we classify the spacelike Dupin hype…
The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.
problem Proving the existence of smooth, entire, strictly convex, spacelike hypersurfaces with constant σk curvature. method Analyzing hypersurfaces in Minkowski space, proving existence through curvature and Gauss map properties.
result Existence of smooth, entire, strictly convex, spacelike hypersurfaces with constant σk curvature. Estimates for spacelike hypersurfaces in de Sitter space.
problem Isometric embedding problem in de Sitter space.
method Obtained Weyl-type estimates and closedness in the space of embeddings.
result Weyl-type estimates and closedness in the space of embeddings.
In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic a…
Compact spacelike hypersurface with constant curvature and boundary angles must be part of a hyperboloid.
problem Characterizing compact spacelike hypersurfaces in Minkowski space with constant curvature and boundary conditions.
method Using an auxiliary function and an associated integral equality, the authors prove the rigidity of the hypersurface.
result Compact spacelike hypersurfaces with constant curvature and boundary angles are rigid, being parts of hyperboloids unless entirely in the boundary hyperplane.
The paper studies a flow of spacelike curves in a Lorentz-Minkowski plane, showing convergence to a constant function.
problem Evolution of spacelike graphic curves in Lorentz-Minkowski plane.
method Anisotropic inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving curves converge to a constant function as time tends to infinity.
The study examines spacelike hypersurfaces in Minkowski space with constant σn−1 curvature.
problem Characterizing spacelike hypersurfaces with constant σn−1 curvature in Minkowski space. method Analyzing hypersurfaces with bounded principal curvatures and proving properties of their convexity.
result Hypersurfaces with constant σn−1 curvature in Minkowski space are either convex or can be split into a product form. Study proves steady state space hypersurfaces are hyperplanes under certain curvature constraints.
problem Characterizing complete spacelike hypersurfaces in steady state space.
method Extended Omori-Yau's maximum principle.
result Proves complete spacelike hypersurfaces are hyperplanes under specific curvature conditions.
In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvatur…
Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.
problem Proving rigidity for hypersurfaces with curvature restrictions.
method Analogue to Guan and Shen's theorem for Riemannian space forms.
result Rigidity theorem for locally isometric hypersurfaces in de Sitter space.
Formula for spacelike submanifolds in warped products.
problem Finding formulas for spacelike submanifolds in semi-Riemannian warped products.
method Extending Simons' type formulas for spacelike submanifolds in semi-Riemannian warped products.
result Compact spacelike hypersurfaces with parallel mean curvature and non-negative sectional curvature are isoparametric hypersurfaces.
Rigidity results for hypersurfaces in warped spacetimes.
problem Characterizing hypersurfaces in specific spacetime geometries.
method Analyzing mean curvature and conformal vectors in warped spacetimes.
result Compact hypersurfaces in certain spacetimes are slices.
The study examines spacelike foliations on Lorentz manifolds under specific conditions.
problem Investigating geometric properties of spacelike foliations on Lorentz manifolds.
method Analyzing conditions for stability, total geodesy, and total umbilicity of foliation leaves.
result Conditions for the stability, total geodesy, and total umbilicity of spacelike foliation leaves are established.
We prove the following stronger verson of the positivity of quasi-local mass stated in gr-qc/0303019: the quasi-local energy (mass) of each connected component of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the space…
Study on the geometry of spacelike hypersurfaces in spacetime.
problem Understanding the geometry of compact spacelike Cauchy hypersurfaces.
method Analysis of a weak Riemannian metric on the manifold of hypersurfaces.
result Positive geodesic distance and non-positive sectional curvature.
Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.
problem Characterizing and understanding spacelike hypersurfaces in twisted product spacetimes.
method Analysis of conditions ensuring global hyperbolicity, use of conformal geometry results.
result Non-existence results for constant mean curvature hypersurfaces under certain conditions.
Study finds unique and non-existent constant mean curvature hypersurfaces in specific spacetimes.
problem Finding unique and non-existent constant mean curvature spacelike hypersurfaces.
method Geometric and physical assumptions applied to Generalized Robertson-Walker spacetimes.
result New uniqueness and non-existence results for complete spacelike hypersurfaces.
We show that a given x:Mn→Mˉn+1=I×φFn closed spacelike hypersurface with constant mean curvature H and warping function φ satisfying φ''\geq \ma\{Hφ'',0\}iseitherminimaloraspacelikeslice\{t_{o}\}\times Fforsomet_{o}\in I$.
We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.
In this paper, we extend a technique due to Romero, Rubio and Salamanca 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 …
Study on compact biconservative hypersurfaces in de Sitter space.
problem Understanding compact biconservative hypersurfaces in de Sitter space.
method Investigation of hypersurfaces with constant scalar curvature under geometric constraints.
result Extension of rigidity properties of biconservative hypersurfaces.
Study finds rigid submanifolds in spacelike waves under specific conditions.
problem Understanding rigidity of submanifolds in spacelike waves.
method Proves submanifolds are contained in characteristic lightlike hypersurfaces under certain conditions.
result Complete codimension two submanifolds are wavefronts under specific conditions.
We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.
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 …
Let Mn be an n-dimensional umbilic-free hypersurface in the (n+1)-dimensional Lorentzian space form M1n+1(c). Three basic invariants of Mn under the conformal transformation group of M1n+1(c) are a 1-form C, called conformal 1-form, a symmetric (0,2) tensor B, called conformal second fun…
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 hy…
We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates.
Anti-de Sitter space is the Lorentzian space form with negative curvature. In this paper we consider lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space with general codimension. In particular, we investigate the singularities of lightlike hypersurfaces as an application of the theory of Legend…
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 hyper…
In this paper we analyze the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete …
We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.
New approach removes obstructions in gluing spacelike and null hypersurfaces in Einstein equations.
problem Gluing two solutions of the Einstein equations along a hypersurface.
method Active utilization of nonlinearity, low-frequency linear analysis, high-frequency nonlinear control.
result Removes 10-dimensional obstructions in null and spacelike gluing problems.
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface Σ≃B(0,1)⊂R3 and the outgoing null hypersurface H emanating from ∂Σ, the time of ex…
Motivated by the important work of Brown adn York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfie…
The paper proves the existence and uniqueness of certain spacelike hypersurfaces with specific curvature and boundary conditions.
problem Existence and uniqueness of convex, entire, spacelike hypersurfaces with constant σk curvature. method Investigation of hypersurfaces with prescribed set of lightlike directions and perturbation on the ideal boundary at infinity.
result Existence and uniqueness of complete entire spacelike constant σk curvature hypersurfaces with prescribed lightlike directions and perturbation. The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.
problem Geometric inequalities for convex hypersurfaces in de Sitter space.
method Locally constrained flows with initial compact spacelike hypersurfaces pinched in de Sitter space.
result Established geometric inequalities related to quermassintegrals and weighted curvature integrals.
The paper solves a conjecture about spacelike hypersurfaces in de Sitter space.
problem Proving an Alexandrov-Fenchel inequality for closed 2-convex spacelike hypersurfaces in de Sitter space.
method Investigating the locally constrained inverse curvature flow to establish the inequality.
result Established an Alexandrov-Fenchel inequality for closed 2-convex spacelike hypersurfaces in de Sitter space.
Proves optimal isoperimetric inequality in de Sitter space.
problem Optimal isoperimetric inequality for specific hypersurfaces in de Sitter space.
method Analyzes spacelike, compact, star-shaped, and 2-convex hypersurfaces in de Sitter space.
result Proves an optimal isoperimetric inequality for the specified hypersurfaces.
In this paper we study the strong stability of spacelike hypersurfaces with constant r-th mean curvature in Generalized Robertson-Walker spacetimes of constant sectional curvature. In particular, we treat the case in which the ambient spacetime is the de Sitter space.
Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
problem Characterize submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
method Analyze light cones, lightlike cylinders, and null cones in specific spacetimes; provide conditions for conformal diffeomorphism.
result Conditions guaranteeing conformal diffeomorphism to hyperbolic space, round cylinder, and sphere.
New findings on hypersurfaces in Euclidean space that are both maximal and minimal.
problem Characterizing hypersurfaces in Euclidean space that are both maximal and minimal.
method Analyzing the level curves of the hypersurfaces and showing they are minimal hypersurfaces in the lower-dimensional Euclidean space.
result The level curves of these hypersurfaces are minimal hypersurfaces in the lower-dimensional Euclidean space.