arXiv research

Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are know…

Paper constructs two series of Lorentz bi-quotients from polyhedra.

Study surfaces with nonvanishing Gauss curvature in Lorentz-Minkowski space.

Classifies zero mean curvature surfaces in Lorentz-Minkowski space.

A condition of Osserman type, called $φ$-null Osserman condition, is introduced and studied in the context of Lorentz globally framed $f$-manifolds. An explicit example shows the naturalness of this condition in the setting of Lorentz $\mathcal{S}$-manifolds. We prove that a Lorentz $\mathcal{S}$-manifold with constant…

Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.

The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.

Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …

The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.

The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie g…

Bipartite Riemann-Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.

A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.

Classified spaces in low dimensions.

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples o…

Study on maximal surfaces with high genus in Lorentz-Minkowski space.

Lecture notes on Lorentz geometry, focusing on curves and surfaces.

The study introduces canonical coordinates for Lorentz surfaces and proves a Bonnet-type theorem.

Investigates differences in solving mean curvature problems in Euclidean and Lorentz-Minkowski spaces.

Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

Unified estimates for mean curvature in Lorentz-Minkowski space.

Study classifies hypersurfaces in 4D Lorentz-Minkowski space using specific operators.

The paper classifies PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.

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

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

Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.

Constructs graphs with singularities in a special space.

The paper studies pseudo-torsion functions of spacelike curves in Lorentz-Minkowski space.

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

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of …

The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.

The paper explores polyharmonic hypersurfaces in pseudo-Riemannian space forms.

Consider a smooth manifold with a smooth metric which changes bilinear type from Riemann to Lorentz on a hypersurface $Σ$ with radical tangent to $Σ$. Two natural bilinear symmetric forms appear there, and we use it to analyze the geometry of $Σ$. We show the way in which these forms control the smooth extensibility ov…

Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.

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

The paper studies loxodromes on twisted surfaces in a specific 3D space.

In this paper, we obtain some properties of biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ having shape operator with complex eigen values. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ whose shape operator has complex eigen values with at most five distinct prin…

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

The paper studies helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space.

Geometric description of Riemann zero mean curvature surfaces in Lorentz-Minkowski space.

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying some curvature conditions…

We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…

New examples of mixed-type zero-curvature graphs found.

A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.

New approach classifies rotational Weingarten surfaces in Lorentz-Minkowski space.

Extends Newton's minimal resistance problem to Lorentz-Minkowski space.

We study the third Laplace-Beltrami operator of timelike rotational surface of (T,L)-type in three dimensional Lorentz-Minkowski space.

LLoCa makes any network Lorentz-equivariant, achieving high accuracy and efficiency.