3-dimensional Lorentz-Minkowski Space Research Articles

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.

Rulings on the Surfaces Having Null Axis and Null Profile Curve in Lorentz-Minkowski 3-Space

We define the curves family of the surfaces with null profile curve and null axis, and give some smooth functions in three dimensional Lorentz-Minkowski space $\mathbb{L}^{3}$. In addition, we compute the third Laplace-Beltrami operator of this type surfaces.

Pogorelov type estimates for a class of Hessian quotient equations in Lorentz-Minkowski space [formula omitted

Let Ω be a bounded domain (with smooth boundary) on the hyperbolic plane Hn(1), of center at origin and radius 1, in the (n+1)-dimensional Lorentz-Minkowski space R1n+1. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of k-convex solutions to a class of Hessian quotient equations defined over Ω⊂Hn(1) and with the vanishing Dirichlet boundary condition.

Application of Matrix Methods On W-Curves

In the present study, we find the parametric equations of non-null W-curves through the semi skew-symmetric matrix in three dimensional Lorentz-Minkowski space. Our technic provides more simple but efficient method for find the parametric equations of these curves in comparison to previous studies in mentioned space. Finally, we give some pictures of W-curves in polynomial form.

Generalized Helices on a Lightlike Cone in 3-Dimensional Lorentz-Minkowski Space

In this paper we provide characterizations and give some properties of generalized helices in 3-dimensional Lorentz-Minkowski space that lie on a lightlike cone. Furthermore, by analyzing their projections, which turn out to be Euclidean or Lorentzian logarithmic spiral, we present their parametrizations. In particular, we also analyze planar generalized helices, that is planar intersections of a lightlike cone.

Behavior of the Gaussian curvature of timelike minimal surfaces with singularities

We prove that the sign of the Gaussian curvature, which is closely related to the diagonalizability of the shape operator, of any timelike minimal surface in the 3-dimensional Lorentz-Minkowski space is determined by the degeneracy and the signs of the two null regular curves that generate the surface. We also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface with some kinds of singular points, which is called a minface. In particular we determine the sign of the Gaussian curvature near any non-degenerate singular point of a minface.

Application of an Inertial Measurement Unit for quantitative assessment of Single Leg Squat in recreational runners

In 3-dimensional Lorentz-Minkowski space we determine the number of catenoids connecting two coaxial circles in parallel planes. This study is separated according to the types of circles and the causal character (spacelike and timelike) of the catenoid.

The number of catenoids connecting two coaxial circles in Lorentz-Minkowski space

In 3-dimensional Lorentz-Minkowski space we determine the number of catenoids connecting two coaxial circles in parallel planes. This study is separated according to the types of circles and the causal character (spacelike and timelike) of the catenoid.

Translation surfaces of some new types in 3-dimensional Minkowski space 𝔼13 satisfying Δri = λiri

Following the publication online, it has been brought to the attention of the Editorial Board of Communications in Contemporary Mathematics (CCM) that the result of this paper was known since 1992 and appeared in the article by Luis Alías, Angel Ferrández and Pascual Lucas, Surfaces in the 3-dimensional Lorentz–Minkowski space satisfying [Formula: see text], Pacific J. Math. 156(2) (1992) 201–208. The information has been verified, and therefore the Editors-in-Chief of CCM have decided to retract the acceptance of this paper.

A sharp height estimate for the spacelike constant mean curvature graph in the Lorentz–Minkowski space

In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness of critical point for the solution of such equation over convex domain, which is an analogue of the result in [28]. Last, by this uniqueness, we obtain a minimum principle for a functional depending on the solution and its gradient. This gives us a sharp gradient estimate for the solution, which leads to a sharp height estimate.

Pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz–Minkowski space

In this paper we introduce the notion of pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz–Minkowski space. We investigate the singularities and geometric properties of these pseudo-spherical normal Darboux images. Furthermore, we investigate the relation of the de Sitter and of the hyperbolic normal Darboux images of a spacelike curve in S 1 2 with the lightlike surface along this spacelike curve.

Convexity of the solutions to the constant mean curvature spacelike surface equation in the Lorentz–Minkowski space

We prove that a spacelike graph of constant mean curvature H≠0 in the 3-dimensional Lorentz–Minkowski space over a bounded domain with pseudo-elliptic boundary is strictly convex. By a pseudo-elliptic curve we mean a closed and planar curve which intersects any branch of any hyperbola at most at five points. We also provide an example that shows that we cannot remove the assumption on the boundary being a pseudo-elliptic curve.

DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE

We review part of the classical theory of curves and surfaces in 3-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.

HELICOIDAL SURFACES IN THE THREE-DIMENSIONAL LORENTZ-MINKOWSKI SPACE SATISFYING ΔIIr = Ar

In this paper, we study helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space under the condition ΔIIri = λiri where ΔII is the Laplace operator with respect to the second fundamental form and λi is a real number. We prove that there are no helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space satisfying that condition.

On quasiconformal harmonic mappings lifting to minimal surfaces

We prove a growth theorem for a function to belong to the class \sum(\mu;a) and generalize a Weierstrass-Enneper representation type theorem for the minimal surfaces given in [5] to spacelike minimal surfaces which lie in 3-dimensional Lorentz-Minkowski space L^3. We also obtain some estimates of the Gaussian curvature of the minimal surfaces in 3-dimensional Euclidean space R^3 and of the spacelike minimal surfaces in L^3.

On the Gauss map of surfaces of revolution in the three-dimensional Minkowski space

In this paper, we study surfaces of revolution without parabolic points in the 3-dimensional Lorentz-Minkowski space whose Gauss map N satisfies the condition ΔIIN = AN, where ΔII is the Laplace operator with respect to the second fundamental form and A is a real 3 × 3 matrix. More precisely we prove that such surfaces are either pseudo-Riemannian spheres S21 or pseudohyperbolic spaces H20.

Pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz–Minkowski space

In this paper we introduce the notion of pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz–Minkowski space which is analogous to the notion of evolutes of curves on the hyperbolic plane. We investigate the singularities and geometric properties of pseudo-spherical evolutes of curves on a spacelike surface.

Space-like surfaces with free boundary in the Lorentz–Minkowski space

We investigate a variational problem in the Lorentz–Minkowski space whose critical points are space-like surfaces with a constant mean curvature and making a constant contact angle with a given support surface along its common boundary. We show that if the support surface is a pseudosphere, then the surface is a planar disc or a hyperbolic cap. We also study the problem of space-like hypersurfaces with free boundary in the higher dimensional Lorentz–Minkowski space .

Helicoidal surfaces in the three-dimensional Lorentz–Minkowski space satisfying Δ II r i = λ i r i

In this paper, we study helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space under the condition Δ II r i = λ i r i where Δ II is the Laplace operator with respect to the second fundamental form and λ i is a real number. We prove that there are no helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space satisfying that condition.

Area minimization among marginally trapped surfaces in Lorentz–Minkowski space

We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz–Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.