arXiv research

Proves isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces.

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.

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and bou…

In this paper we generalize in Lorentz-Minkowski space $ł^3$ the two-dimensional analogue of the catenary of Euclidean space. We solve the Dirichlet problem for bounded mean convex domains and spacelike boundary data that have a spacelike extension to the domain. We also classify all singular maximal surfaces of $ł^3$ …

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.

Unique CMC foliation in Minkowski space solved.

The paper estimates curvature for specific hypersurfaces in a special space.

We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space $\mathbb{R}^{n,m}$, which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of…

Paper proves isoperimetric inequality for Minkowski spacetime.

The paper examines critical points of solutions to a surface equation in 3D spacelike spaces.

Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for …

Spacelike surfaces in the Lorentz-Minkowski space L^3 can be endowed with two different Riemannian metrics, the metric inherited from L^3 and the one induced by the Euclidean metric of R^3. It is well known that the only surfaces with zero mean curvature with respect to both metrics are open pieces of the helicoid and …

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness of cr…

New geometric approach gives apriori estimate for optimal transport maps.

A normal field on a spacelike surface in $R^4_1$ is called bi-normal if $K^ν$, 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.

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

In this paper, we give the characterizations of spacelike B2-slant helix by means of curvatures of the spacelike curve in Minkowski 4-space. Furthermore, we give the integral characterization of the spacelike B2-slant helix.

Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian Möbius strips of non constant curvature all of whose spacelike geodesics are cl…

In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space $\mathbb{R}^{m+n}_{n}$. Firstly, we classify $m$-dimensional complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ by affine technique and classical…

In this paper, using the classifications of timelike and spacelike ruled surfaces, we define and study the Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space. We give the conditions for spacelike offset surfaces to be developable.

Estimates for spacelike hypersurfaces in de Sitter space.

In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed spacelike curve with a spacelike binormal in dual Lorentzian space.

Proves inequality for maximal spacelike submanifolds in Minkowski space.

The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.

The paper proves rigidity for self-shrinking spacelike graphs in pseudo-Euclidean space.

Study of spacelike submanifolds in spherical RW spacetime, proving a Lorentzian Takahashi theorem.

Study proves spacelike self-shrinkers are hyperplanes under certain conditions.

Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.

Study finds rigid submanifolds in spacelike waves under specific conditions.

Proves stability of Minkowski space for specific initial data.

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…

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…

In this paper, we give some characterizations for spacelike helices in Minkowski space-time. We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space-time.

The paper studies geometric flows of spacelike curves in Lorentz-Minkowski plane and proves their long-term behavior.

To study spacelike surfaces in the Lorentz-Minkowski space $\Bbb R^{4}_1,$ we construct a pair of maps whose values are in the lightcone, called $\mathfrak l_r^{\pm}$-Gauss maps. We can use these maps to study umbilical spacelike surfaces and find parametrizations of spacelike surfaces of revolution of hyperbolic and …

Let $Ω$ be a compact and mean-convex domain with smooth boundary $Σ:=\partialΩ$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $Σ$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $Σ$ admits an isometric and isospin immersion…

On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must b…

In this paper, position vector of a spacelike general helix with respect to standard frame in Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike general helix. In terms of solution, …

New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.

We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observation…

Compact spacelike hypersurface with constant curvature and boundary angles must be part of a hyperboloid.

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternio…

Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular …

The study examines spacelike hypersurfaces in Minkowski space with constant $σ_{n-1}$ curvature.

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…