Lorentz Minkowski Space Research Articles

Pseudo-circular evolutes and involutes of lightcone framed curves in the Lorentz–Minkowski 3-space

In this paper, we first define the pseudo-circular evolute of a lightcone framed curve in the Lorentz–Minkowski 3-space, and show that the pseudo-circular evolute of a lightcone framed curve can be seen as a generalization of the circular evolute of a framed curve without type changing. We also define an involute of a lightcone framed curve, and study the duality relations between involutes and pseudo-circular evolutes. Moreover, we study the pseudo-circular evolutes and involutes with respect to a lightcone circle frame, and investigate the relationship with respect to a projection map. Meanwhile, examples are given for explaining the objects we investigated, which also show a certain physical significance of our investigation.

Geometry of surfaces with singular points—applications to maximal surfaces in Lorentz-Minkowski space

Geometry of surfaces with singular points—applications to maximal surfaces in Lorentz-Minkowski space

Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space

Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space

Timelike zero mean curvature surfaces in ℝ1 4

Abstract We are interested in the solution of the Björling problem for timelike surfaces in R 1 4 \mathbb{R}_{1}^{4} . The main contribution of the paper is to present new and many examples of timelike zero mean curvature surfaces and give their explicit parametric equations. In particular cases, one observes that the parametric equations of these surfaces coincide with the timelike minimal surfaces in Lorentz–Minkowski 3-space.

A non‐Newtonian perspective on multiplicative Lorentz–Minkowski space 𝕃∗3

This study focuses on redesigned the Lorentz–Minkowski space, great importance in both physics and geometry, by employing multiplicative metric concepts (angle, norm, distance etc.). The impact of multiplicative arguments on the Lorentz‐Minkowski space has been elucidated, showcasing how multiplicative vectors, subspaces, and planes exhibit multiplicative spacelike, timelike, or lightlike characteristics. Also, we conducted comprehensive comparisons with conventional scenarios, elucidating the vectors and their characteristics within this distinct space. To facilitate visual comprehension, illustrative examples and figures have been provided, offering a clearer understanding of the subject matter.

A new glimpse into the directional curves associated with a pseudo-null curve via Bishop frame

In this study, the directional associated curves of a pseudo-null curve have been investigated according to Bishop frame in Lorentz-Minkowski 3-space. The relational equations have been obtained between the curvature functions of the pair donor-associated curves and the distance function. Furthermore, an answer has been sought to the question: does a self-associated curve of a pseudo-null curve via Bishop frame exist or not in Lorentz-Minkowski 3-space?

A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions

AbstractWe develop a conformal duality for space‐like graphs in Riemannian and Lorentzian three‐manifolds that admit a Riemannian submersion over a Riemannian surface whose fibers are the integral curves of a Killing vector field, which is time‐like in the Lorentzian case. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz–Minkowski space with prescribed mean curvature a bounded function with bounded gradient. On the other hand, we obtain conditions for the existence and nonexistence of entire graphs which are related to a notion of the critical mean curvature.

Umbilics of Surfaces in the Lorentz–Minkowski 3-Space

In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz–Minkowski space {{mathbb {L}}}^3. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in {{mathbb {L}}}^3 is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer m, there exists a germ of a space-like surface with an isolated C^{infty }-umbilic (resp. C^1-umbilic) of index (3-m)/2 (resp. 1+m/2). We also show that the indices of isolated umbilics of time-like surfaces in {{mathbb {L}}}^3 that are not the accumulation points of quasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.

Curvature estimates for spacelike graphic hypersurfaces in Lorentz–Minkowski space R1n+1$\mathbb {R}^{n+1}_{1}$

AbstractIn this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the ‐dimensional Lorentz–Minkowski space , and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed 2‐nd Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane of center at origin and radius 1, can be proven.

Classification of Ruled Surfaces as Homothetic Self-Similar Solutions of the Inverse Mean Curvature Flow in the Lorentz–Minkowski 3-Space

Classification of Ruled Surfaces as Homothetic Self-Similar Solutions of the Inverse Mean Curvature Flow in the Lorentz–Minkowski 3-Space

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb R}^{n+1}_1$

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb R}^{n+1}_1$

Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are translating solitons for the inverse mean curvature flow. In particular, we prove the existence of non-cylindrical ruled translating solitons, which contrast with the Euclidean setting.

On the harmonic evolute of time-like Hasimoto surfaces in Lorentz–Minkowski space

The movement of a thin vortex in a thin viscous fluid by the motion of a curve propagating in Lorentz–Minkowski space [Formula: see text] is described by the vortex filament or smoke ring equation and can be viewed as a dynamical system on the space curves in [Formula: see text]. This paper investigates the harmonic evolute surfaces of time-like Hasimoto surfaces in [Formula: see text]. Also, we discuss the geometric properties of these surfaces, namely, we obtain the Gaussian and mean curvatures of the first and second fundamental forms. As a verification, we construct a concrete example for the meant surfaces to demonstrate our theoretical results.

Higher genus nonorientable maximal surfaces in the Lorentz-Minkowski 3-space

We study nonorientable maximal surfaces in Lorentz-Minkowski 3-space. We prove some existence results for surfaces of this kind with high genus and one end.

On extended biharmonic hypersurfaces with three curvatures

The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics and differential geometry, arises from the variation problems of ordinary mean curvature vector field. Generally, harmonic submanifolds are biharmonic, but not vice versa. Of course, many examples of biharmonic hypersurfaces are harmonic. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every biharmonic submanifold is harmonic. Although the conjecture has not been proven (in general case), it has been affirmed in many cases, and this has led to its spread to various types of submanifolds. Inspired by the conjecture, we study the Lorentz submanifolds of the Lorentz-Minkowski spaces. We consider an advanced versión of the conjecture (namely, L1-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean 5-space L5 := E15 (i.e. the Minkowski 5-space). We confirm the extended conjecture on Lorentz hypersurfaces with three principal curvatures.

On reaction-diffusion models with memory and mean curvature-type diffusion

We propose and study a reaction-diffusion model with memory, where the linear diffusion operator is replaced by the mean curvature operator both in Euclidean and Lorentz–Minkowski spaces and the memory kernel is assumed to be of Jeffreys type. Regarding the reaction term, we consider a balanced bistable function f=−F′, with F a generic double well potential with wells of equal depth. In particular, we assume that the potential F has two global minima at ±1 and that F(u)∼|1±u|2+θ, for some θ>−1, when u≈±1, and we consider the corresponding equation in a bounded interval with homogeneous Neumann boundary conditions. We prove that if θ∈(−1,0), then there exist special steady states, named compactons, with a transition layer structure. In contrast, if θ≥0 the interface layers are not stationary and two different phenomena emerge: for θ=0 solutions exhibit a metastable behavior and maintain an unstable structure for an exponentially long time, while if θ>0 the exponentially slow motion is replaced by an algebraic one.

On the Gaussian curvature of timelike surfaces in Lorentz-Minkowski 3-space

In this study, the various expressions of the Gaussian curvature of timelike surfaces whose parameter curves intersect under any angle are investigated and the Enneper formula is obtained in Lorentz-Minkowski 3-space. By giving an example for these surfaces, the graphs of the surface and its Gaussian curvature are drawn.

Statistical structures arising in null submanifolds

We show a link between affine differential geometry and null submanifolds in a semi-Riemannian manifold via statistical structures. Once a rigging for a null submanifold is fixed, we can construct a semi-Riemannian metric on it. This metric and the induced connection constitute a statistical structure on the null submanifold in some cases. We study the statistical structures arising in this way. We also construct statistical structures on a null hypersurface in the Lorentz–Minkowski space using the null second fundamental form. This extends the classical construction to the null case.

Generating Matrices of Rotations in Minkowski Spaces using the Lie Derivative

This paper aims to generate materices of rotations in Minkowski using the Lie Derivative. The calculus on manifolds in Lorentzian spaces are used to generate matrices of rotation in three-dimensional Lorentz-Minkowski space which includes one axis in timelike and the other two are spacelike axes. The findings showed that the manifolds and their calculus dramatically increased the use of Lie derivative in many branches of mathematics and physics, The findings also revealed that matrices ( of rotation) leave one line ( axis) fixed and these matrices of rotation are used widely in differential geometry in physics. Furthermore, the findings demonstrated that any surfaces of revolution inside this space must be invariant under one of these matrices. The main result of this paper is a new procedure of creating rotational matrices explicitly using the Lie derivative and deriving it into a linear system of ordinary differential equaion. Solving this system leads to matrices of rotation that leaves one axis fixed in Minkowski space.

Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space

In this paper, we define surfaces of revolution without parabolic points in three-dimensional Lorentz–Minkowski space. Then, we classify this class of surfaces under the condition ΔIIIx=Ax, where ΔIII is the Laplace operator regarding the third fundamental form, and A is a real square matrix of order 3. We prove that such surfaces are either catenoids or surfaces of Enneper, or pseudo spheres or hyperbolic spaces centered at the origin.