广义de Sitter空间中的类时超曲面的切触性质

从切触几何及Legendrian奇点理论的角度研究了广义de sitter空间中的类时超曲面的切触性质及gdS-高斯像的奇点的分类和几何意义.     

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.     

奇异积分算子在 H~1(R_+~2×R_+~2)上的有界性

本文用 H~1(R_+~2×R_+~2)函数原子分解的方法,证明由 R.Fefferman 及 E.M.Stein 在[1]、[2]中引进的乘积空间上的奇异积分算子,当核函数 K(x_1,x_2)满足的条件中的 η>1/2时,在 H~2(R_+~2×R_+~2)上是有界的.即存在与 f 无关的常数 C,使‖K*f‖_H~1≤C‖f‖_(H~1).     

de Sitter空间中具平行平均曲率向量的完备类空子流形(Ⅱ)

本文证明了de Sitter空间中具平行平均曲率向量的完备类空子流形在其第二基本形式模长平方S＜2(n-1c)的平方根时是全脐子流形。     

Morse指数与带权的椭圆方程的Liouville型定理

该文研究全空间R~N上带权的半线性椭圆型方程-△u=|x|~α|u|~(p-1)u,x∈R~N与半空间R_+~N={x∈R~N:x_N0}上带权的半线性椭圆型问题-△u=|x|~α|u|~(p-1)u,x∈R_+~N,u|?R_+~N=0的Liouville型定理,其中N≥3,α-2.证明了,当1p(N+2α+2)/(N-2)时,上述问题的Morse指数有限的有界解只能是零解.     

de Sitter空间中法联络平坦的完备类空子流形

该文证明了de Sitter空间中具有平行平均曲率向量的常数量曲率完备类空子流形,如果其法联络是平坦的,且M的截面曲率小于0,或M的第二基本形式模长平方‖σ‖相似文献   

R_1~3中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R_(1)~3曲面的等距变形问题.建立了R_(1)~3中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R_(1)~3出现的新情况——曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

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.     

对数核Toader平均的Schur凸性和Schur几何凸性

为了研究对数核Toader平均L_r(a,b)在R_(++)~2上的Schur凸性和Schur几何凸性,利用控制不等式的相关理论得到结论:当r≥1/2时,L_r(a,b)在R_(++)~2上是Schur凹函数;当r≥0时,L_r(a,b)在R_(++)~2上是Schur几何凸函数;当r≤0时,L_r(a,b)在R_(++)~2上是Schur几何凹函数,最后,依据L_r(a,b)的Schur凸性和Schur几何凸性建立了新的不等式.     

常曲率黎曼流形的极小子流形

本文目的在于建立下述定理:常曲率 a 的黎曼流形 V~(n p)中的紧致无边极小子流形M~n 常满足∫_(Mn){p∑R_(ijkl)~2 2p∑R_(ij)~2-R~2 n(3p-2n 2)aR}*1≥n~2(n-1)(n-p-1)a~2Vol(M~n),其中∑R_(ijkl)~2是M~n 的黎曼曲率张量的模长平方,∑R_(ij)~2是 M~n 的李齐(Ricci)曲率张量的模长平方,R 是 M~n 的数量曲率.上述积分不等式是 M~n 的内在性质.     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

Unique continuation from infinity in asymptotically anti-de Sitter spacetimes II: Non-static boundaries

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations □_g?+σ? = 𝒢(?,??) on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting nonstatic boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudo-convex hypersurfaces near the conformal boundary.     

Special relativity and theory of gravity via maximum symmetry and localization

Like Euclid,Riemann and Lobachevski geometries are on an almost equal footing,based on the principle of relativity of maximum symmetry proposed by Professor Lu Qikeng and the postulate on invariant universal constants c and R,the de Sitter/anti-de Sitter(dS/AdS)special relativity on dS/AdS-space with radius R can be set up on an almost equal footing with Einstein's special relativity on the Minkowski-space in the case of R→∞. Thus the dS-space is coin-like:a law of inertia in Beltrami atlas with Beltrami time simultaneity for the principle of relativity on one side,and the proper-time simultaneity and a Robertson-Walker-like dS-space with entropy and an accelerated expanding S~3 fitting the cosmological principle on another side. If our universe is asymptotic to the Robertson-Walker-like dS-space of R(?)(3/Λ)~(1/2),it should be slightly closed in O(A)with entropy bound S(?)3πc~3k_B/ΛGh.Contrarily,via its asymptotic behavior, it can fix on Beltrami inertial frames without‘an argument in a circle’and acts as the origin of inertia. There is a triality of conformal extensions of three kinds of special relativity and their null physics on the projective boundary of a 5-d AdS-space,a null cone modulo projective equivalence[N](?)_p(AdS~5). Thus there should be a dS-space on the boundary of S~5×AdS~5 as a vacuum of supergravity. In the light of Einstein's‘Galilean regions’,gravity should be based on the localized principle of relativity of full maximum symmetry with a gauge-like dynamics.Thus,this may lead to the theory of gravity of corresponding local symmetry.A simple model of dS-gravity characterized by a dimensionless constant g(?)(AGh/3c~3)~(1/2)～10~(-61)shows the features on umbilical manifolds of local dS-invariance. Some gravitational effects out of general relativity may play a role as dark matter. The dark universe and its asymptotic behavior may already indicate that the dS special relativity and dS-gravity be the foundation of large scale physics.     

The singularities of null surfaces in Anti de Sitter 3-space

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.     

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     

三维De Sitter与Anti-de Sitter空间中的等参曲面

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.     

Notes on a paper of Mess

These notes are a companion to the preceding paper by Geoffrey Mess, “Lorentz spacetimes of constant curvature”. Mess’ paper was written nearly 20 years ago and so we hope these notes will be useful as a guide to the literature that has appeared in the intervening years. Lars Andersson was supported in part by the NSF, contract no. DMS 0407732 and Thierry Barbot was supported by CNRS, ACI “Structures géométriques et Trous Noirs”.     

Integral formulas for compact spacelike hypersurfaces in de sitter space and their applications to Goddard's conjecture

We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter spaceS ₁ ⁿ⁺¹ (1) and give their applications to the case of constantr-th mean curvature (r=1,2,…,n−1). Whenr=1 we recover Montiel's result. Li Haizhong is supported by NNSFC No.19701017 and Basic Science Research Foundation of Tsinghua University and Chen Weihua is supported by NNSFC No. 19571005