arXiv research

The paper proves rigidity theorems for space-like λ-hypersurfaces in Lorentzian space.

Study classifies space-like hypersurfaces in de Sitter space with parallel tensors.

The paper classifies space-like hypersurfaces with parallel Blaschke tensors in de Sitter space.

Study classifies space-like hypersurfaces with parallel Blaschke tensor in conformal space.

We classify the space-like biharmonic surfaces in 3-dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space.

The paper studies how certain surfaces evolve in space-time.

The paper proves rigidity theorems for space-like hypersurfaces in Minkowski space.

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space $H^{n+1}_1(-1)$. we prove that if a complete space-like hypersurface with constant mean curvature $x:\mathbf M\rightarrow H^{n+1}_1(-1)$ has two distinct principal curvatures $λ,μ$, and inf$|λ-μ|>0$, then $x$ is the sta…

In this paper we introduce the fourth fundamental form for the hypersurfaces in $H^{n+1}$ and the space-like hypersurfaces in $S_{1}^{n+1}$ and discuss the conformality of the normal Gauss maps of the hypersurfaces in $H^{n+1}$ and $S_{1}^{n+1}$. Particularly, we discuss the surfaces with conformal normal Gauss maps in…

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal sectional curvatures are locally equivalent to canal hypersurfaces in Euclidean spa…

There are considered 4-dimensional pseudo-Riemannian spaces with inner products of signature (3,1) and (2,2). The objects of investigation are space-like and time-like hyperspheres in the respective cases. These hypersurfaces are equipped with almost contact B-metric structures. The constructed manifolds are characteri…

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter $N$-dimensional s…

Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.

We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is the boun…

On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions…

Study of GCR hypersurfaces in Minkowski spaces, including classification and examples.

Study of potential Carroll structures and special Carrollian manifolds for null hypersurfaces.

The paper studies biharmonic submanifolds in pseudo-Riemannian manifolds.

Study of complete space-like self-expanders in Minkovski space.

Study proves Euler characteristic of certain space-like manifolds is positive.

The paper studies Gauss maps of space-like stationary surfaces in Lorentz-Minkowski space, focusing on ramification and unicity.

Study space-like surfaces in Robertson-Walker spacetimes with specific geometric conditions.

The paper provides Weierstrass representations for minimal space-like surfaces in Minkowski space-time.

On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as …

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and introducing space-like height function on the unit speed time-like curves on $\mathbb{S}^{2}_{1}$, the invariants of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ and geometric properties of de Si…

Improves Bernstein theorem for space-like graphs in Lorentz-Minkowski space.

The paper proves a stability result for translating space-like graphs in Lorentz manifolds.

Solves PDE system for minimal space-like surfaces in Minkowski space-time.

The paper provides a Weierstrass representation for maximal space-like surfaces in 4D pseudo-Euclidean space.

The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.

Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.

When the vacuum Einstein equations are cast in the form of hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold MΣ of riemannian metrics on a Cauchy hypersurface Σ. As in every lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those …

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-…

Study space-like loxodromes on canal surfaces in Minkowski 3-space.

In this work we firstly classify space-like surfaces in Minkowski space $\mathbb E^4_1$, de-Sitter space $\mathbb S^3_1$ and hyperbolic space $\mathbb H^3$ with harmonic Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of the first kind. We also give s…

Research explores the space-like embeddings in pseudo-hyperbolic space, finding geometric frames and actions.

A complete characterization is obtained of the asymptotic behavior of solutions of the static vacuum Einstein equations which have a (pseudo)-compact horizon or boundary and are complete away from the boundary. It is proved that the time-symmetric space-like hypersurface has only finitely many ends, each of which is ei…

The conformal geometry of surfaces in the conformal space $\mathbf Q^n_1$ is studied. We classify the space-like surfaces in $\mathbf Q^n_1$ with vanishing conformal form up to conformal equivalence.

We prove some Bernstein theorems for entire space-like submanifolds in pseudo-Euclidean spaces and, as a corollary, we obtain a new proof of the Calabi-Pogorelov theorem on global solutions of Monge-Ampere equations.

Paper finds space-like maximal surfaces with entire null lines in 3D space-time.

Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.

Proves existence of AdS manifold with prescribed metrics on boundary.

We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominat energy condition. The appropriate existence and regularity results are established in the special case o…

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null…

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been…

The paper studies umbilics on surfaces in Lorentz-Minkowski space.

We study space-like self-shrinkers of dimension $n$ in pseudo-Euclidean space $\ir{m+n}_m$with index $m$. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove a rigidity results under minor growth conditions interms of the mean curv…

Method provides solutions to constraint equations for Lorentzian manifolds with lightlike parallel spinors.