Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Spacelike Hypersurfaces with Constant Scalar Curvature in the Lorentz–Minkowski Space

We show that the only compact spacelike hypersurfaces in the Lorentz–Minkowski space Lⁿ⁺¹ having nonzero constant scalar curvature and spherical boundary are the hyperbolic caps (with negative constant scalar curvature). One key ingredient in our proof will be an integral formula for the n-dimensional volume enclosed by the boundary of a compact spacelike hypersurface, in the case where the boundary is contained in a hyperplane of Lⁿ⁺¹. As a direct application of that integral formula we also derive an interesting result for the volume of spacelike hypersurfaces.     

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.     

Closed Strong Spacelike Curves,Fenchel Theorem and Plateau Problem in the 3-Dimensional Minkowski Space

The authors generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here the strong spacelike condition means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve γ under consideration. The assumption of index 1 is equivalent to saying that γ winds around some timelike axis with winding number 1. This reversed Fenchel-type inequality is proved by constructing a ruled spacelike surface with the given curve as boundary and applying the Gauss-Bonnet formula. As a by-product, this shows the existence of a maximal surface with γ as the boundary.     

Position vectors of spacelike general helices in Minkowski 3-space

In this paper, the position vectors of a spacelike general helix with respect to the standard frame in Minkowski space are studied in terms of the Frenet equations. First, a vector differential equation of third order is constructed to determine the position vectors of an arbitrary spacelike general helix. In terms of solution, we determine the parametric representation of the general helices from the intrinsic equations. Moreover, we give some examples to illustrate how to find the position vectors of spacelike general helices with a spacelike and timelike principal normal vector.     

三维Minkowski空间中曲面的广义Weierstrass公式

本文给出了三维Ｍｉｎｋｏｗｓｋｉ空间中一般类空曲面与类时曲面的广义Ｗｅｉｅｒ－ｓｔｒａｓｓ表示公式．     

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.     

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.     

Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II

In this sequel paper, we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient conditions of divergence type for the monotonicity of the Hawking mass. These flows of surfaces may have connections to the problem in general relativity of bounding the total mass of a spacetime from below by the quasi-local mass of spacelike 2-surfaces in the spacetime.     

On the Ricci Curvature of Compact Spacelike Hypersurfaces in de Sitter Space

In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for the Ricci curvature. Our result will be a consequence of an integral formula involving the Ricci curvature and the scalar curvature of the hypersurface. We also derive some other consequences and applications of this formula.     

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.     

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R2,1. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU₂ with SU1,1. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form SU1,1, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R2,1. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.     

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.     

A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds

In this paper we use the standard formula for the Laplacian of the squared norm of the second fundamental form and the asymptotic maximum principle of H. Omori and S.T. Yau to classify complete CMC spacelike hypersurfaces of a Lorentz ambient space of nonnegative constant sectional curvature, under appropriate bounds on the scalar curvature.     

Constant Mean Curvature Surfaces Foliated by Circles in Lorentz–Minkowski Space

We prove that a spacelike surface in L3 with nonzero constant mean curvature and foliated by pieces of circles in spacelike planes is a surface of revolution. When the planes containing the circles are timelike or null, examples of nonrotational constant mean curvature surfaces constructed by circles are presented. Finally, we prove that a nonzero constant mean curvature spacelike surface foliated by pieces of circles in parallel planes is a surface of revolution.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.     

n维Anti-de Sitter空间中类空超曲面的类时Gauss像的奇点

本文利用奇点理论研究了n维Anti-deSitter空间中的类空超曲面,介绍了类空超曲面的局部微分几何,定义了类时Anti-deSitterGauss像及Anti-deSitter高度函数,并进一步的利用Anti-deSitter高度函数族和流形间的切触理论研究了类时Anti-deSitterGauss像的几何意义及类空超曲面的通有性.最后研究了类空超曲面的AdS-Monge型.     

An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space

We describe a robust method for constructing a tubular surface surrounding a spacelike curve with a spacelike principal normal in Minkowski 3-Space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface’s skin at points where the curve experiences high torsion. In our construction the tubular surface’s twist is bounded by the spacelike curve’s curvature and is independent of the spacelike curve’s torsion.      

Born–Infeld solitons,maximal surfaces,and Ramanujan’s identities

We show that a Born–Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born–Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan’s identities and the Weierstrass–Enneper representation of maximal surfaces, we derive further non-trivial identities.     

Lightlike surfaces of Darboux-like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3-space

In this paper, by introducing a new frame on spacelike curves lying in lightcone 3-space, we investigate the geometric properties of the lightlike surface of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix generated by spacelike curves in lightcone 3-space. As an extension of our previous work and an application of the singularity theory, the singularities of the lightlike surfaces of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix are classified, several new invariants of spacelike curves are discovered to be useful for characterizing these singularities, meanwhile, it is found that the new invariants also measure the order of contact between spacelike curves or principal normal indicatrixes of spacelike curves located in lightcone 3-space and two-dimensional lightcone whose vertices are at the singularities of lightlike surfaces. One concrete example is provided to illustrate our results.