arXiv research

Study minimal time-like surfaces in 4D space-time, proving conditions for existence.

Canonical coordinates defined for minimal time-like surfaces in n-dimensional Minkowski space.

We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface 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 no…

Study links Hopf differentials to curvature line flows on time-like CMC surfaces.

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

The study defines invariants for time-like surfaces with real asymptotic lines.

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…

The paper classifies time-like surfaces in a static space-time.

The paper extends a maximal surface to a time-like surface in Lorentz-Minkowski space.

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

Study on surfaces in neutral space forms with zero mean curvature.

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

Study Born-Infeld solitons and solve Björling problem for them.

The paper studies sections of time-like twistor spaces with specific covariant derivatives.

We relate in this note the classical Schwarzian derivative to the curvature of time-like curves in Lorentz surfaces of constant curvature.

Two self-similar solutions found for time-like hypersurfaces in Minkowski spacetime.

We study the neutral Kähler metric on the space of time-like lines in Lorentzian ${\Bbb{E}}^3_1$, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metr…

No global solutions found for time-like minimal submanifolds in Minkowski space.

Penrose limit results for specific 3-surfaces in space-time.

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a gen…

The paper examines surfaces in 3D space-times with bounded mean curvature.

The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.

Investigates physical properties on surfaces of rotation using Clairaut's theorem.

Study variational problem for time-like curves in Einstein universe.

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $π$. We interpret this result in terms of Teichmüller theory, and prove the existence of …

Study classifies translating solitons in Minkowski 3-space, revealing singularities.

Study loxodromes and geodesics on rotational surfaces in pseudo-isotropic space.

Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…

In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E$^3_1$ satisfies a vector differential equation of fourth order. The general so…

Study of generalized Bishop frames on time-like curves in 4D Lorentz space.

The paper characterizes surfaces in 4D space forms with flat normal connection.

New proof shows minimal surfaces globally exist for compact data.

The paper analyzes equations for surfaces in 4D space forms.

The paper extends foliation by constant mean curvature surfaces to spacetimes with particles.

We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows mor…

We prove two related results. The first is an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $π$: any two such metrics in are connected by a unique left earthquake. The second result is that the space of ``globally hyperbolic'' AdS manifolds with ``parti…

Paper proves a new criterion for time-like geodesics in flat spacetimes.

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…

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

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 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…

Unified estimates for mean curvature in Lorentz-Minkowski space.

The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.

In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannia…

We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities of angles less than $2π$ along a time-like graph $Γ$. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient …

Study on spectrum of Dirac operator on pseudo-Riemannian manifolds.

The parametrization theorem is derived in a flat nD pseudo-complex affine space. The pseudo-complex hyperbolic space accomodates n-number of uncompactified time-like extra dimensions with sugnature (s,r), where s and r are the numbers of minus and plus signs associated with the diagonalized metric matrix. The main resu…

Researchers prove unique foliation of AdS space-time with constant Gauss curvature surfaces.