广义de Sitter空间中法向Lorentzian超曲面的nullcone Gauss映射的奇点

利用奇点理论研究了广义de Sitter空间中具有Lorentzian法空间的一类超曲面.介绍了这类超曲面的局部微分几何,定义了nullcone Gauss映射及nullcone高度函数族,进而研究了nullcone高度函数族的性质及nullcone高斯映射的几何意义,最后研究了这类超曲面的通有性质.     

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.     

指标为2的半四维欧氏空间中三维Lorentzian子流形的Anti de Sitter高斯映射的奇点

作为R_(2~4)空间中子流形研究的补充工作,试图利用Lagrange奇点理论对R_(2~4)空间中三维Lorentzian子流形的微分几何性质及奇异性进行研究.     

On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space

In this paper we prove that a complete spacelike hypersurface in de Sitter space such that its image under the Gauss map is contained in a hyperbolic geodesic ball of radius is necessarily compact and its -dimensional volume satisfies , where denotes the volume of a unitary round -sphere. We also characterize the case where these inequalities become equalities. As an application of our result, we also conclude that Goddard's conjecture is true under the assumption that the hyperbolic image of the hypersurface is bounded.

     

New examples of maximal space like surfaces in the anti-de Sitter space

Let be the three-dimensional anti-de Sitter space. In this paper we will construct new examples of complete maximal space like surfaces in . Moreover, we will prove that any complete maximal space like surface in with principal curvatures ±κ bounded away from zero must be isometric to the hyperbolic cylinder. Since the new examples that we have constructed have exactly two principal curvatures everywhere, we conclude that the condition on the principal curvatures on the previous result, i.e. the condition |κ(m)|>c>0, cannot be replaced by the condition |κ(m)|>0.     

Singularities of de Sitter Gauss map of timelike hypersurface in Minkowski 4-space

We study the singularities of de Sitter Gauss map of timelike hypersurface in Minkowski 4-space through their contact with hyperplanes.     

Nontrivial Soliton Scattering in 2+1 Anti-de Sitter Space

Using analytic methods, we present integrable solutions of the Bogomolny Yang–Mills–Higgs equations in 2+1 anti-de Sitter space. In particular, families of soliton solutions are constructed explicitly and their dynamics is investigated in some detail.     

On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S⁴ and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to P ³.Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.     

Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ3 1(−1)

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ?³ ₁(?1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ?SL₂? via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E ³ ₁. The hyperbolic Gauß map of timelike surfaces in ?³ ₁(?1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ?³ ₁(?1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ?³ ₁(?1) is studied.     

在Sn1+1空间中具有常数量曲率的类空超曲面的高斯映射

在这篇文章中,我们研究在de Sitter空间中具有非负常值的第r个平均曲率的紧致的类空超曲面.我们证明了在合适的条件下紧致的类空超曲面是全脐的.     

Complete spacelike hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we investigate the complete spacelike hypersurfaces with constant mean curvature and two distinct principal curvatures in an anti-de Sitter space. We give a characterization of hyperbolic cylinder and prove the conjecture in a paper by L. F. Cao and G. X. Wei [J. Math. Anal. Appl., 2007, 329(1): 408–414].     

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.     

On the Gauss map of B-scrolls in 3-dimensional Lorentzian space forms

In this note we show that B-scrolls over null curves in a 3-dimensional Lorentzian space form are characterized as the only ruled surfaces with null rulings whose Gauss maps G satisfy the condition being a parallel endomorphism of .     

An integral inequality for compact maximal surfaces inn-dimensional de Sitter space and its applications

In this paper we prove an integral inequality for the Gaussian curvature of compact maximal surfaces inn-dimensional de Sitter space. Some applications of that inequality are given in order to solve the associated Bernstein type problem as well as to characterize the totally geodesic immersion in terms of its area and the first nontrivial eigenvalue of its Laplacian.Partially supported by a DGICYT Grant No. PB91-0705-C02-02Partially supported by a DGICYT Grant No. PB91-0731     

3维双曲空间中曲面的双曲Gauss映照和法Gauss映照

本文导出了3维双曲空间中曲面的双曲Gauss映照和法Gauss映照的关系,发现了一般的曲面由双曲Gauss映照和平均曲率函数唯一确定,并证明了双曲Gauss映照所满足的二阶线性椭圆方程,给出了两种形式的关于双曲Gauss映照的三阶非线性偏微分方程(组)的一个解.     

Immersed surfaces of prescribed Gauss curvature into Minkowski space

Given a positive real valued function on the disc, we will immerse the disc into three dimensional Minkowski space in such a way that Gauss curvature at the image point of is . Our approach lies on the construction of Gauss map of surfaces.

     

anti-de Sitter 空间中紧致类空超曲面的积分公式及其在常高阶平均曲率下的应用

该文对 anti-de Sitter 空间H₁ⁿ⁺¹中的紧致类空超曲面建立了积分公式,并应用它们在常高阶平均曲率的条件下讨论了H₁ⁿ⁺¹中紧致类空超曲面的全脐问题.     

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

Minkowski空间中保高斯映射的共形形变

设Ｍｎ为Ｒｉｅｍａｎｎ流形，给定类空浸入：Ｍｎ→Ｒｎ，ｐ，如果存在另一个类空浸入：Ｍｎ→Ｒｎ，ｐ，使与在共形对应之下且对应点的地空间平行，则称类空子流形是可保高斯映射共形形变的．本文给出可保高斯映射共形形变的充要条件．对ｎ＝２，ｐ＝１的情形，如果上述形变是同向的，我们分类了曲面；如果是反向的，我们用主曲率满足的方程来描述．     

广义de Sitter空间中的类时超曲面

利用奇点理论研究广义de Sitter空间中的类时超曲面.介绍类时超曲面的局部微分几何,定义了广义de Sitter高斯像及广义de Sitter高度函数,研究广义deSitter高度函数族的性质及广义de Sitter高斯像的几何意义,介绍了一种证明高度函数为Morse族的新方法.最后研究了类时超曲面的通有性质.