共查询到20条相似文献,搜索用时 609 毫秒
1.
Donghe Pei 《Applicable analysis》2013,92(6):1165-1180
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. 相似文献
2.
Conformal geometry of surfaces in Lorentzian space forms 总被引:4,自引:0,他引:4
We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Dedicated to Professor T.J. WillmoreSupported by an FPPI Postdoctoral Grant from DGICYT Ministerio de Educación y Ciencia, Spain 1994 and by a DGICYT Grant No. PB94-0750-C02-02 相似文献
3.
Qiyu Chen Jean-Marc Schlenker 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(1):181-216
We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”. 相似文献
4.
Oscar M. Perdomo 《Geometriae Dedicata》2011,152(1):183-196
In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows
that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two
algebraic maximal hypersurfaces in the anti de Sitter N-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature hypersurfaces
in the de Sitter space. 相似文献
5.
A spacelike surface M in 3-dimensional de sitter space S13 or 3-dimensional anti-de Sitter space H13 is called isoparametric, if M has constant principal curvatures. A timelike surface is called isoparametric, if its minimal polynomial of the shape operator is constant. In this paper, we determine the spacelike isoparametric surfaces and the timelike isoparametric surfaces in S13 and H13. 相似文献
6.
The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory
of relativity. In this paper we consider spacelike curves in de Sitter 3-space. We define the notion of lightlike surfaces
of spacelike curves in de Sitter 3-space. We investigate the geometric meanings of the singularities of such surfaces.
Work partially supported by Grant-in-Aid for formation of COE. ‘Mathematics of Nonlinear Structure via Singularities’ 相似文献
7.
In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map. 相似文献
8.
We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory. We define two mappings associated
to a timelike surface which are called Anti de Sitter nullcone Gauss image and Anti de Sitter torus Gauss map. We also define a family of functions named Anti de Sitter null height function on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities
of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map. 相似文献
9.
Liang Chen Qixing Han Weizhi Sun 《Journal of Mathematical Analysis and Applications》2010,366(1):256-265
We study the geometric properties of degenerate surfaces, which are called AdS null surfaces, in Anti de Sitter 3-space from a contact viewpoint. These surfaces are associated to spacelike curves in Anti de Sitter 3-space. We define a map which is called the torus Gauss image. We also define two families of functions and use them to investigate the singularities of AdS null surfaces and torus Gauss images as applications of singularity theory of functions. 相似文献
10.
In this paper, we show that a complete spacelike hypersurface immersed with constant mean curvature either in the de Sitter space or in the anti-de Sitter space must be totally umbilical, provided that its Gauss mapping has some suitable behavior. In particular, we use an extension of Hopf’s maximum principle due to Yau in order to give new positive answers to Goddard’s conjecture. 相似文献
11.
We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the
physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by
the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like
surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic
manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal
surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized
by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space
of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional
gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which
are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between
hyperbolic surfaces, that allows an extension to cone-manifolds.
相似文献
12.
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. 相似文献
13.
We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from
the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in
AdS
n+1, any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface.
In AdS3, if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature.
As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional
projective geometry. 相似文献
14.
A. V. Barabanschikov S. F. Prokushkin M. A. Vasiliev 《Theoretical and Mathematical Physics》1997,110(3):295-304
We reformulate the free equations of motion for massive spin-0 and spin-1/2 matter fields in (2+1)-dimensional anti-de Sitter
space in the form of some constant curvature conditions. The infinite-dimensional representation of the anti-de Sitter algebra,
which is basic to this formulation, admits a natural realization in terms of the algebra of deformed “oscillators” with a
deformation parameter related to the mass parameter.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 372–384, March, 1997. 相似文献
15.
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. 相似文献
16.
Shyuichi Izumiya Donghe Pei María del Carmen Romero Fuster 《Proceedings of the Steklov Institute of Mathematics》2009,267(1):156-173
We define the notions of (S
t
1 × S
s
2)-nullcone Legendrian Gauss maps and S
+2-nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between
singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian
singularities. By using S
+2-nullcone Lagrangian Gauss maps, we define the notion of S
+2-nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion
of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence,
we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space,
hyperbolic space, Lorentz-Minkowski space and de Sitter space. 相似文献
17.
3维双曲空间中曲面的双曲Gauss映照和法Gauss映照 总被引:3,自引:0,他引:3
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. 相似文献
18.
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. 相似文献
19.
Bang-Yen Chen 《Central European Journal of Mathematics》2009,7(3):400-428
Submanifolds with parallel mean curvature vector play important roles in differential geometry, theory of harmonic maps as
well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean curvature vector were classified by B.
Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean
spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean curvature vector in pseudo-Riemannian
spheres and pseudo-hyperbolic spaces with arbitrary codimension and arbitrary index. Consequently, we achieve the complete
classification of spatial surfaces with parallel mean curvature vector in all pseudo-Riemannian space forms. As an immediate
by-product, we obtain the complete classifications of spatial surfaces with parallel mean curvature vector in arbitrary Lorentzian
space forms.
相似文献
20.
We prove that every three-dimensional maximal globally hyperbolic spacetime, locally modelled on the anti-de Sitter space
AdS
3, with closed orientable Cauchy surfaces, admits a unique CMC time function. 相似文献