Lorentz Minkowski Research Articles

Pedal Curves of the Mixed-Type Curves in the Lorentz-Minkowski Plane

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane R12. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve.

Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given.

Area Maximizing Surfaces in Lorentzian Spaces

In this paper we provide new results for area maximizing compact spacelike surfaces with boundary embedded in Lorentz–Minkowski space, as well as establish the uniqueness of the Dirichlet problem for maximal graphs in the aforementioned space. Moreover, we extend our results to more general Lorentzian spaces that admit an infinitesimal timelike symmetry.

Isophote curves on spacelike surfaces in Lorentz–Minkowski space

An isophote curve consists of a locus of surface points whose normal vectors make a constant angle with a fixed vector (the axis). In this paper, we define an isophote curve on a spacelike surface in Lorentz–Minkowski space [Formula: see text] and then find its axis as timelike and spacelike vectors via the Darboux frame. Besides, we give some relations between isophote curves and special curves on surfaces such as geodesic curves, asymptotic curves or lines of curvature.

Accretive Darboux growth in Lorentz–Minkowski spacetime

It is known that the geometric methods are used in most fields in natural sciences. Kinematics on curves and surfaces as one of these methods is an essential tool for investigating the growth of some biological objects. In this study, the time‐dependent and quaternionic models of accretive growth are considered. For this, first, it is defined as a growth velocity in the direction of the Darboux vector at every point on a spatial non‐null curve in three‐dimensional Lorentz–Minkowski spacetime. Second, accretive surface growth is presented through quaternions. With the help of the quaternionic form of the growth model, we reach that a point (cell tract) on the accretive surface makes a homothetic motion. Also, several examples and visualizations are given to support the theory.

On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz–Minkowski spacetime 𝕃m

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

Spacelike Surfaces in [formula omitted] with null mean curvature vector and the nonlinear Riccati partial differential equation

This paper introduces a complex Weierstrass representation for weakly conformal spacelike immersions in the Lorentz–Minkowski space L4 involving holomorphic or anti-holomorphic functions. Factoring through the light cone, we parametrize these immersions by three complex functions (μ,a,b). We prove the existence of a correspondence between those immersions for which a is holomorphic and b satisfies a Riccati partial differential equation, with conformal immersions in the hyperbolic space H3 with lightlike mean curvature vector. As a principal application of this correspondence, we obtain a powerful tool to construct new, explicit solutions of nonlinear Riccati partial differential equation and their explicit, associated surfaces. We also use our complex representation to classify the totally umbilic, weakly conformal immersions in L4, when a and b are both anti-holomorphic functions.

A Note on Spacelike Submanifolds Through Light Cones in Lorentzian Space Forms

We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones $$\Lambda _{c}(p)$$ of de Sitter and anti-de Sitter spacetimes. Every light cone $$\Lambda _{c}(p)$$ contains the lightlike geodesics starting from p and, essentially, it coincides with the horisms $$E^{+}(p)\cup E^{-}(p)$$ . The analysis works by means of an explicit correspondence with the spacelike submanifolds through the light cone in the Lorentz–Minkowski spacetime. In particular, a characterization of totally umbilical compact surfaces through light cones in de Sitter and anti-de Sitter is shown and we obtain an estimation of the first eigenvalue of the Laplace operator on a compact spacelike surface in a light cone.

Evolutes of the (n,m)-cusp mixed-type curves in the Lorentz–Minkowski plane

The evolute of a regular curve in the Lorentz–Minkowski plane is given by the locus of centers of osculating pseudo-circle of the base curve. But the case when a curve has singularities is not very clear. In this paper, we use lightcone frame to define the [Formula: see text]-cusp mixed-type curves and their evolutes in Lorentz–Minkowski plane. In order to attain this goal, we define the [Formula: see text]-cusp non-lightlike curves and their evolutes in Lorentz–Minkowski plane first. Then we study the behaviors of the evolutes of the [Formula: see text]-cusp mixed-type curves at the [Formula: see text]-cusp.

Rigidity results of free boundary maximal hypersurfaces in the region bounded by a de Sitter space

This paper provides the rigidity results of free boundary maximal hypersurfaces in a region bounded by a de Sitter space in the Lorentz–Minkowski space. First, it is proved that any smooth, compact free boundary maximal hypersurface in a de Sitter ball is the spacelike coordinate planar disk passing through the center of the de Sitter space. Second, a smooth, noncompact, complete free boundary maximal hypersurface in the exterior of a de Sitter ball is considered. For maximal hypersurfaces with one planar end in the exterior of a de Sitter ball, every complete noncompact free boundary maximal hypersurface is shown to be a part of the spacelike coordinate hyperplane.

Classification of f-Biharmonic Curves in Lorentz–Minkowski Space

In this paper, we firstly derived the equations for the curves of a Lorentz–Minkowski space L3 to be f-biharmonic. Then, using these equations, we classify such unit speed curves in L3.

Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion

Reaction-diffusion equations are widely used to describe a variety of phenomena such as pattern formation and front propagation in biological, chemical and physical systems. In the one-dimensional model with a balanced bistable reaction function, it is well-known that there is persistence of metastable patterns for an exponentially long time, i.e. a time proportional to exp⁡(C/ε) where C,ε are strictly positive constants and ε2 is the diffusion coefficient. In this paper, we extend such results to the case when the linear diffusion flux is substituted by the mean curvature operator both in Euclidean and Lorentz–Minkowski spaces. More precisely, for both models, we prove existence of metastable states which maintain a transition layer structure for an exponentially long time and we show that the speed of the layers is exponentially small. Numerical simulations, which confirm the analytical results, are also provided.

The Lorentzian Version of a Theorem of Krust

In Lorentz–Minkowski space, we prove that the conjugate surface of a maximal graph over a convex domain is also a graph. We provide three proofs of this result that show a suitable correspondence between maximal surfaces in Lorentz–Minkowski space and minimal surfaces in Euclidean space.

Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space

Abstract The isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space 𝐄 1 3 {\mathbf{E}_{1}^{3}} and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane ℒ 2 {\mathcal{L}^{2}} ). In this paper, we shall give necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in 𝐄 1 3 {\mathbf{E}_{1}^{3}} . We shall discuss the causal character of the second derivative of unit speed spacelike curves in 𝐄 1 3 {\mathbf{E}_{1}^{3}} in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.

On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating diffusions (monotone and non monotone) and singular nonlinear diffusions are considered. In particular, the cases of mean curvature-type diffusions both in the Euclidean space and in Lorentz–Minkowski space enter in our framework. After dealing with existence and stability of monotone steady states in a bounded interval of the real line with Dirichlet boundary conditions, we discuss the speed rate of convergence to the asymptotic limit as t→+∞. Finally, in the particular case of a Burgers flux function, we show that the solutions exhibit the phenomenon of metastability.

Bernstein-Type Theorem for Zero Mean Curvature Hypersurfaces Without Time-like Points in Lorentz–Minkowski Space

Calabi and Cheng-Yau’s Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz–Minkowski $$(n+1)$$ -space $${{\varvec{R}}_{1}^{n+1}}$$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like points. Using this, we give an improvement of the Bernstein-type theorem, and we show that an entire zero mean curvature graph in $${\varvec{R}}^{n+1}_1$$ consisting only of space-like or light-like points is a hyperplane. This is a generalization of the first, third and fourth authors’ previous result for $$n=2$$ .

A New Angular Measurement in Minkowski 3-Space

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples.

Null helices and Cartan slant helices in Lorentz–Minkowski 3-space

In this paper, we study null helices, Cartan slant helices and two special developable surfaces associated to them in Lorentz–Minkowski 3-space. We give a method using a special plane curve to construct a null helix. We also define the null tangential Darboux developable of a null Cartan curve, and we give a classification of singularities of it. Moreover, we study the relationship between null helices (or Cartan slant helices) with the developable surfaces of them.

Spacelike Zero Mean Curvature Surfaces in $${\mathbb {L}}^{4}$$

We present new explicit parametrizations of spacelike zero mean curvature surfaces in four-dimensional Lorentz–Minkowski space $${\mathbb {L}}^{4}$$. The surfaces are the solutions of the Bjorling problem whose core curve is a circle and a helix and the tangent planes are expressed in terms of trigonometric functions.

Spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space

We investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space Ln+1, for n≥2, spanning a given boundary datum lying on the hyperbolic space ℍn.