3-dimensional Lorentz-Minkowski Space Research Articles

Isometric Generalized Maximal Surfaces in Lorentz–Minkowski Space

We show that if two generalized non-planar maximal immersions of a connected Riemann surface into three dimensional Lorentz–Minkowski space are isometric, then they must be associate, up to a congruence of the ambient space. The corresponding result for minimal surfaces is due to H. Schwarz.

Stationary bands in three-dimensional Minkowski space

In this paper we consider a free boundary problem in the 3-dimensional Lorentz-Minkowski space L 3 which deals spacelike surfaces whose mean curvature is a linear function of the time coordinate and the boundary moves in a given support plane. We study spacelike surfaces that project one-to-one into a strip of the support and that locally are critical points of a certain energy functional involving the area of the surface, a timelike potential and preserves the volume enclosed by the surface. We call these surfaces stationary bands. We establish existence of such surfaces and we investigate their qualitative properties. Finally, we give estimates of its size in terms of the initial data.

A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz–Minkowski space and application

In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σ n immersed in the ( n + 1 ) -dimensional Lorentz–Minkowski space L n + 1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of L n + 1 . Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of L n + 1 .

An analytic extension of a spacelike maximal surface

It is shown that a spacelike maximal surface in the three dimensional Lorentz-Minkowski space can be extended analytically if it meets a spacelike plane at a constant hyperbolic angle.

Spacelike Hypersurfaces with Free Boundary in the Minkowski Space under the Effect of a Timelike Potential

In this paper we consider a variational problem for spacelike hypersurfaces in the (n + 1)-dimensional Lorentz-Minkowski space \(\mathbb{L}^{n+1}\), whose critical points are hypersurfaces supported in a spacelike hyperplane Π determined by two facts: the mean curvature is a linear function of the distance to Π and the hypersurface makes a constant angle with Π along its boundary. We prove that the hypersurface is rotational symmetric with respect to a straight-line orthogonal to Π and that each (non-empty) intersection with a parallel hyperplane to Π is a round (n − 1)-sphere. A similar result is proved for hypersurfaces trapped between two parallel hyperplanes.

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space

We prove that a complete embedded maximal surface in = (ℝ3, dx12 + dx22-dx32) with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid. We show that the moduli space of entire maximal graphs over {x3=0} in with n+1≥2 singular points and vertical limit normal vector at infinity is a 3n+4-dimensional differentiable manifold. The convergence in means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x3=0}. Moreover, the position of the singular points in ℝ3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space of marked graphs with n+1 singular points (a mark in a graph is an ordering of its singularities), which is a (n+1)-sheeted covering of . We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space is an analytic manifold of dimension 3n−1. This manifold can be identified with a spinorial bundle associated to the moduli space of Weierstrass data of graphs in .

Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying $$ \Delta ^{{II}} \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} = A\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} $$

In this paper the surfaces of revolution without parabolic points, in the 3-dimensional Lorentz-Minkowski space are classified under the condition $$ \Delta ^{{II}} \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} = A\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} $$ where Δ II 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 minimal or Lorentz hyperbolic cylinders or pseudospheres of real or imaginary radius.

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz–Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz–Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.

Constant mean curvature spacelike hypersurfaces with spherical boundary in the Lorentz-Minkowski space

We study compact spacelike hypersurfaces (necessarily with non-empty boundary) with constant mean curvature in the ( n + 1)-dimensional Lorentz-Minkowski space. In particular, when the boundary is a round sphere we prove that the only such hypersurfaces are the hyperplanar round balls (with zero mean curvature) and the hyperbolic caps (with non-zero constant mean curvature).

On the Gauss map of $B$-scrolls

B-scrolls over null curves in the 3-dimensional Lorentz-Minkowski space L are characterized as the only ruled surfaces with null rulings whose Gauss map satisfies the condition ∆G = ΛG, Λ being and endomorphism of L. This note completes the classification of such surfaces given by S.M. Choi in Tsukuba J. Math. 19 (1995), 285–304.

Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying Δx=Ax+B

In this paper we locally classify the surfaces M s 2 in the 3-dimensional Lorentz-Minkowski space L 3 verifying the equation Δx = Ax + B, where A is an endomorphism of L 3 and B is a constant vector. We obtain that classification by proving that M s 2 has constant mean curvature and in a second step we deduce M s 2 is isoparametric

On surfaces in the 3-dimensional Lorentz-Minkowski space

Let M s be a surface in the 3-dimensional Lorentz-Minkowski space L 3 and denote by H its mean curvature vector field. This paper locally classifies those surfaces verifying the condition ∆H = λH , where λ is a real constant. The classification is done by proving that M s has zero mean curvature everywhere or it is isoparametric, i.e., its shape operator has constant minimal polynomial.