Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space

For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space, one can naturally introduce two Gauss maps and a Weierstrass-type representation. In this paper we investigate the global geometry of such surfaces systematically. The total Gaussian curvature is related with the surface topology as well as the indices of the so-called good singular ends by a Gauss–Bonnet type formula. On the other hand, as shown by a family of counterexamples to Osserman?s theorem, finite total curvature no longer implies that Gauss maps extend to the ends. Interesting examples include the deformations of the classical catenoid, the helicoid, the Enneper surface, and Jorge–Meeks? k-noids. Each family of these generalizations includes embedded examples in the 4-dimensional Lorentz space, showing a sharp contrast with the 3-dimensional case.     

On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We deal with complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, having two distinct principal curvatures. In this setting, we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space, under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form. Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with some generalized maximum principles.     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

Rotational linear Weingarten surfaces of hyperbolic type

A linear Weingarten surface in Euclidean space ℝ³ is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c ∈ ℝ. Such a surface is said to be hyperbolic when a ² + 4bc < 0. In this paper we study rotational linear Weingarten surfaces of hyperbolic type giving a classification under suitable hypothesis. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in ℝ³ that consists of surfaces with self-intersections whose generating curves are periodic. Partially supported by MEC-FEDER grant no. MTM2007-61775.     

Ruled Weingarten surfaces in the Galilean space

In this paper we study ruled Weingarten surfaces in the Galilean space. Weingarten surfaces are surfaces having a nontrivial funcional relation between their Gaussian and mean curvature. We describe some further examples of Weingarten surfaces.      

Complete Linear Weingarten Spacelike Hypersurfaces Immersed in a Locally Symmetric Lorentz Space

In this paper, we establish a characterization theorem concerning the complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, whose sectional curvature is supposed to obey certain appropriated conditions. Under a suitable restriction on the length of the second fundamental form, we prove that a such spacelike hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.     

Spacelike submanifolds of codimension two in anti-de Sitter space

In this paper, we study spacelike submanifolds of codimension two in anti-de Sitter space from the viewpoint of Legendrian singularity theory. We introduce the notion of the anti-de Sitter normalized Gauss map which is a generalization of the ordinary notion of Gauss map of hypersurfaces in Euclidean space. We also introduce the AdS-normalized Gauss–Kronecker curvature for a spacelike submanifold of codimention two in anti-de Sitter space. In the local sense, this curvature describes the contact of submanifolds with some model surfaces.     

Discrete maximal surfaces with singularities in Minkowski space

We describe discrete maximal surfaces with singularities in 3-dimensional Minkowski space and give a Weierstrass type representation for them. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space generally have certain singularities. We give a criterion that naturally describes the “singular set” for discrete maximal surfaces, including a classification of the various types of singularities that are possible in the discrete case.     

New Characterizations of Compact Totally Umbilical Spacelike Surfaces in 4-Dimensional Lorentz–Minkowski Spacetime Through a Lightcone

On each spacelike surface through the lightcone in 4-dimensional Lorentz–Minkowski spacetime, there exists an Artinian normal frame which contains the position vector field. In this way, a (globally defined) lightlike normal vector field, with nontrivial extrinsic meaning, is chosen on the surface. When the second fundamental form respect to that normal direction is non-degenerate, a new formula which relates the Gauss curvature of the induced metric and the Gauss curvature of this normal metric is obtained. Then, the totally umbilical round spheres are characterized as the only compact spacelike surfaces through the lightcone whose normal metric has constant Gauss curvature two. Such surfaces are also distinguished in terms of the Gauss–Kronecker curvature of that lightlike normal direction, of the area of the normal metric and of the first non-trivial eigenvalue of the Laplacian of the induced metric.     

A Remark on Bcklund Transformation of Weingarten Surfaces

50.Intr0ducti0nInLlj,weestablishedacorrespondencebetweensolutionsofSine-Gordonequati0nandWeingartensurfacesinR3withc0ndition'K-2mH+m2+l2=O,l#0,(0'l)andgavetheBdcklundtransformationofthiskindofWeingartensurfacesintermsofso-calledDarbouxcongruence-Itisremarkablethatthisthe0ryissimilartotheclassicalthe0ryofBacklundtransformationandpseudo-sphericalcongruencebetweensurfacesofnegativeconstantGausscurvature.Inthispaper,wewanttorelatethemt0eachotherbyso-calledparalleltransformation.51.ParallelTr…     

Lorentz空间中常平均曲率类空超曲面

本文证明了ｎ＋１维Ｌｏｒｅｎｔｚ空Ｌｎ＋１中以Ｓｎ－１（ｒ）为边界的紧致常平均曲率类空超曲面只有 Ｂｎ（ｒ）和超伪球面盖．对于 Ｒｎ＋１中的紧致常平均曲率超曲面，当高斯映照像落在一个半球面内时，得到相应的唯一性结果．     

Lie geometry of linear Weingarten surfaces

We show how linear Weingarten surfaces appear as special Ω-surfaces and give a characterization of those linear Weingarten surfaces that allow a Weierstrass type representation.     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

Ruled Weingarten surfaces in Minkowski 3-space

We characterize all ruled surfaces in Minkowski 3-space with a relation between the Gauss and mean curvature (Weingarten surfaces). It turns out that, except if the rulings are in a null direction, these are given by Lorentzian screw motions of straight lines. However, if the rulings are always in a null direction, then every ruled surface is Weingarten. Received: 9 February 1998 / Revised version: 20 December 1998     

Critical curves for the total normal curvature in surfaces of 3-dimensional space forms

A variational problem closely related to the bending energy of curves contained in surfaces of real 3-dimensional space forms is considered. We seek curves in a surface which are critical for the total normal curvature energy (and its generalizations). We start by deriving the first variation formula and the corresponding Euler–Lagrange equations of these energies and apply them to study critical special curves (geodesics, asymptotic lines, lines of curvature) on surfaces. Then, we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear Weingarten surface. Finally, we give some classification and existence results for this family of rotation surfaces.     

Invariant surfaces in the homogeneous space Sol with constant curvature

A surface in homogeneous space is said to be an invariant surface if it is invariant under some of the two 1‐parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.     

Linear Weingarten Spacelike Submanifolds in Semi-Riemannian Space Form

Mathematical Notes - We consider the linear Weingarten spacelike submanifold immersed in semi-Riemannian space form $$\mathbb {N}^{n+p} _q (c)$$ of constant sectional curvature $$c$$ and index...     

An exterior boundary value problem in Minkowski space

In three‐dimensional Lorentz–Minkowski space ??³, we consider a spacelike plane Π and a round disc Ω over Π. In this article we seek the shapes of unbounded surfaces whose boundary is ? Ω and its mean curvature is a linear function of the distance to Π. These surfaces, called stationary surfaces, are solutions of a variational problem and governed by the Young–Laplace equation. In this sense, they generalize the surfaces with constant mean curvature in ??³. We shall describe all axially symmetric unbounded stationary surfaces with special attention in the case that the surface is asymptotic to Π at the infinity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Translation surfaces of constant curvatures in a simply isotropic space

In this paper we describe, up to a congruence, translation surfaces in a simply isotropic space having constant isotropic Gaussian or mean curvature. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature \(K\ne 0\) and translation surfaces with constant mean curvature \(H\ne 0\) that are not cylindrical. Furthermore, we investigate a class of Weingarten translation surfaces.