Non-positively curved surfaces can be embedded in flat spacetime.
problem Embedding surfaces with non-positive curvature in flat spacetime.
method Polyhedral approximation to prove isometric embedding.
result Metric of non-positive curvature can be embedded as a convex spacelike Cauchy surface.
Solves a Cauchy problem for minimal spacelike surfaces in 4D spacetime.
problem Constructing minimal spacelike surfaces in 4D spacetime.
method Defining isoclinic parametric surfaces and proving their relation to holomorphic functions, solving the Cauchy problem.
result Solves the Cauchy problem for minimal spacelike surfaces in R24. New method constructs spacelike data leading to trapped surfaces.
problem Formation of trapped surfaces from spacelike initial data.
method Free data formalism and local existence result.
result Data can be extended to asymptotically flat Cauchy data.
Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.
problem Analyzing the relationship between Lorentz harmonic maps and spacelike surfaces.
method Using loop group techniques, develop DPW-type representations and solve Cauchy problems.
result Establish a correspondence between Lorentz harmonic maps and spacelike immersions, leading to families of surfaces of constant Gauss curvature.
When studying the causal propagation of a field in a globally hyperbolic spacetime M, one often wants to express the physical intuition that it has compact support in spacelike directions, or that its support is a spacelike compact set. We compare a number of logically distinct formulations of this idea, and of the com…
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.
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 …
We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence D in (2+1)-dimensional Minkowski space, provided D is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the …
Researchers create initial data for multiple collapsing boson stars.
problem Forming multiple trapped surfaces in spacetime.
method Constructed Cauchy initial data for EMKG system.
result Multiple trapped surfaces form in finite time.
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.
No trapped surfaces can form under low-regularity bounds in certain spacetimes.
problem Existence of trapped surfaces in low regularity solutions to Einstein's equations.
method Analyzing the initial data in Besov B2,13/2 norm and extending to H3/2 smallness. result No trapped surfaces can exist initially when the Cauchy data are close to Minkowski spacetime data.
Paper discusses existence of CMC surfaces in cosmological spacetimes.
problem Existence of constant mean curvature (CMC) Cauchy surfaces in cosmological spacetimes.
method Review of existing results and discussion of connections to spacetime splitting.
result Connection between CMC surfaces and spacetime splitting problem.
Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.
problem Embedding cone-metrics in anti-de Sitter spacetimes.
method Proving embeddings using Fuchsian representations and GHMC spacetimes.
result Unique embeddings of cone-metrics in GHMC anti-de Sitter spacetimes.
Proves that timelike maximal surfaces in (1+2) Minkowski space are always embedded.
problem Characterizing timelike maximal surfaces in (1+2) Minkowski space.
method Analyzing smooth proper timelike immersions with vanishing mean curvature.
result Timelike maximal surfaces are always embedded, with compact subsets being smooth graphs.
Paper studies Einstein vacuum equations with low regularity data.
problem Einstein vacuum equations with low regularity initial data.
method Combines Klainerman-Szeftel-Rodnianski curvature theorem, Czimek's extension procedure, and global elliptic estimates.
result Time of existence controlled by low regularity bounds on curvature in L2. Authors review CMC spacelike hypersurfaces and new existence results.
problem Existence of constant mean curvature (CMC) spacelike hypersurfaces in cosmological spacetimes.
method Review and new existence results based on previous work and conjectures.
result New existence results for CMC spacelike hypersurfaces.
Study of maximal surfaces in a specific Heisenberg group with singularities.
problem Characterize maximal surfaces in the Lorentzian Heisenberg group with singularities.
method Use harmonic maps into the 2-sphere, loop group construction, and solve the Cauchy problem.
result Regular maximal discs must have at least two cuspidal cross-cap singularities on the boundary.
Study on 3D spacetimes, focusing on vacuum data and energy bounds.
problem Existence and properties of solutions for Einstein equations in 3D spacetimes.
method Analysis of 2D general relativistic initial data sets, construction of vacuum, spacelike data, and review of global Hamiltonian charges.
result Established lower bounds for energy in terms of angular momentum, linear momentum, and center of mass.
New results on non-existence and rigidity of spacelike submanifolds in spacetimes.
problem Non-existence and rigidity of spacelike submanifolds with causal mean curvature vector field.
method General results for various spacetimes including globally hyperbolic, stationary, and pp-wave spacetimes.
result Significant consequences in Geometrical Analysis, solving new Calabi-Bernstein and Dirichlet problems.
Given a globally hyperbolic spacetime M, we show the existence of a {\em smooth spacelike} Cauchy hypersurface S and, thus, a global diffeomorphism between M and R×S.
We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.
Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.
problem Embedding flat metrics on hyperbolic surfaces into (2+1)-spacetimes.
method Using convex polyhedral Cauchy surfaces and Teichmüller space properties.
result Existence and uniqueness of flat metrics embedding in (2+1)-spacetimes.
The paper extends Hawking--Page solutions to various spacetimes with singularities.
problem Understanding the extensions of Hawking--Page solutions with different types of singularities.
method Kaluza--Klein reduction and Christodoulou's methods.
result Extensions of Lorentzian Hawking--Page solutions with null, spacelike singularities, and Cauchy horizons of Taub--NUT type are proven.
Solves the Cauchy problem for linearised Einstein equation on globally hyperbolic spacetimes.
problem Initial value problem for gravitational waves on globally hyperbolic vacuum spacetimes.
method Proves the solution map is an isomorphism of locally convex topological vector spaces and solves linearised constraint equations on closed manifolds.
result Well-posedness of the Cauchy problem for gravitational waves on globally hyperbolic spacetimes.
The paper strengthens a singularity theorem in General Relativity.
problem Proving conditions under which spacetime is incomplete or has specific geometric structures.
method Improving a previous theorem by Galloway and Ling, the paper introduces new conditions for spacetime properties.
result Conditions for spacetime to be past null geodesically incomplete, or have specific geometric structures.
Study flat spacetimes with BTZ singularities using BTZ-extension.
problem Understanding singularities in flat spacetimes.
method Use BTZ-extension to describe Moduli spaces and construct cauchy-surface.
result Construct convex polyhedral cauchy-surface in Cauchy-compact flat spacetimes with BTZ.
Proves well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
problem Proving well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
method Analyzes globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces.
result Proves well-posedness of the Cauchy problem for the Dirac operator.
We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemann…
The paper studies surfaces in Minkowski space with constant curvature.
problem Properly embedded spacelike surfaces in Minkowski 3-space with constant Gaussian curvature.
method Classification of surfaces with bounded prescribed Gaussian curvature, proving uniqueness of foliations.
result Every regular domain not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature.
The paper solves a generalized catenary problem in Lorentz-Minkowski space.
problem Solving the Dirichlet problem for specific geometric domains.
method Generalization of catenary to Lorentz-Minkowski space and solution of Dirichlet problem.
result Classification and solution of singular maximal surfaces.
New black hole models with both null and spacelike singularities.
problem Understanding singularities in black hole spacetimes.
method Developed a new spacelike-characteristic gluing method to construct black hole spacetimes.
result First examples of black holes with coexisting null and spacelike singularities.
Study shows no closed trapped submanifolds can be tangent to certain spacelike hypersurfaces.
problem Existence of closed trapped submanifolds in spacetime regions foliated by specific hypersurfaces.
method Introduced k−future convex spacelike/null hypersurfaces and proved no k−dimensional closed trapped submanifolds can be tangent to these hypersurfaces from their future side. result Closed trapped submanifolds cannot be found in open spacetime regions foliated by k−future convex hypersurfaces. 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.
Unique AdS spacetime found with prescribed metric on a convex surface.
problem Finding an AdS spacetime with a specific metric on a convex surface.
method Constructing a quasifuchsian AdS spacetime with a past-convex Cauchy surface.
result Existence and uniqueness of a quasifuchsian AdS spacetime with the specified properties.
Proves stability of Minkowski space for specific initial data.
problem Stability of Minkowski space under spacelike-characteristic initial data.
method Vectorfield method and bootstrapping argument, with new geometric constructions.
result Global nonlinear stability of Minkowski space proved for the spacelike-characteristic Cauchy problem.
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 study connects curves in Anti-de Sitter space to hyperbolic plane diffeomorphisms.
problem Understanding the geometry of curves in Anti-de Sitter space.
method Using a correspondence between spacelike surfaces and area-preserving diffeomorphisms of the hyperbolic plane.
result Every quasisymmetric homeomorphism of the circle admits a unique extension as a θ-landslide of the hyperbolic plane.
In this paper, we study entire spacelike translating solitons in Minkowski space. By constructing convex spacelike solutions to (1.3) in bounded convex domains, we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity.
We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely …
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 explores conditions for homothetic Killing vectors on spacetime hypersurfaces.
problem Conditions for the existence of homothetic Killing vectors on spacetime hypersurfaces.
method General identities relating deformation tensor and tensor on hypersurfaces, applied to specific settings.
result Necessary and sufficient conditions for homothetic Killing vectors on spacetime hypersurfaces.
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. The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.
problem Mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.
method Analysis of natural curvature pinching and noncollapsing quantities under mean curvature flow.
result The mean curvature flow deforms any initial spacelike-convex submanifold to a point in finite time, and is asymptotic to a shrinking sphere in a maximally spacelike subspace.
We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in…
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. A normal field on a spacelike surface in R14 is called bi-normal if Kν, the determinant of Weingarten map associated with ν, is zero. In this paper we give a relationship between the spacelike pseudo-planar surfaces and spacelike pseudo-umbilical surfaces, then study the bi-normal fields on spacelike ruled sur…
In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim …
Study spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.
problem Characterize spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.
method Find parametrizations of spacelike loxodromes on both spacelike and timelike helicoidal surfaces.
result Classification of spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.