Remarks on Complete Spacelike Hyper-surfaces with Constant Mean Curvature in the de Sitter Space

Let M be an n(≥3)-dimensional completely non-compact spacelike hypersurface in the de Sitter space S1 (n+1) (1) with constant mean curvature and non negative sectional curvature. It is proved that M is isometric to a hyperbolic cylinder or an Euclidean space if H ≥1. When 2(n-1)~(1.2)/n < H < 1, there exists a complete rotation hypersurfaces which is not a hyperbolic cylinder.     

SURFACE FINITE ELEMENTS FOR

In this article we define a surface finite element method （SFEM） for the numerical solution of parabolic partial differential equations on hypersurfaces F in R^n＋1. The key idea is based on the approximation of F by a polyhedral surface Гh consisting of a union of simplices （triangles for n = 2, intervals for n = 1） with vertices on Г. A finite element space of functions is then defined by taking the continuous functions on Гh which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on Г. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demorrstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.     

李奇曲率平行的黎曼流形的曲率张量模长

李安民和赵国松［１］提出了下面的问题：找出李奇曲率平行的黎曼流形的曲率张量模长的最佳拼挤常数并确定达到该值的流形．本文确定了非爱因斯坦流形的最佳拼挤常数和达到该值的黎曼流形．在ｎ１２时，回答了［１］中提出的问题．     

一类具有平行平均曲率和有限全曲率的完备子流形

本文把Berard P.,do Carmo M.,Santos W.在1998年所得的结果,分别推广到局部对称的Cartan-Hadamard流形中具有常平均曲率和有限全曲率的完备超曲面,以及球面上具有平行平均曲率和有限全曲率的完备子流形.     

复射影空间CP^n＋p的Kaehler子流形

本文对复射影空间ＣＰｎ＋ｐ中具有非负全实双截面曲率的ｎ维（ｐ＜ｎ）紧致Ｋａｅｈｌｅｒ子流形作了完全分类．     

半经典Brans-Dicke理论中的闭合宇宙解

在计及真空涨落的半经典Brans-Dicke理论中，找到了一个小尺度因子时的闭合宇宙解.它存在一个初始类空的曲率奇点. 关键词：     

具有平行平均曲率向量场的子流形

本文改进了Ｓ．Ｔ．Ｙａｕ（文［１］）中关于单位球面中具有平行平均曲率向量场的子流形的一个结果。然而从截面曲率这一角度出发，给出了空间形式Ｒ＾ｎ＋ｐ（ｃ）（ｎ＞１，ｐ＞１）中具有平行平均曲率向量场的可定向闭子流形Ｍ＾ｎ的有关结果和积分不等式。     

广义非线性模型的影响分析

本文从几何的观点研究广义非线性模型及其影响分析，我们给出了均值漂移模型的曲率度量；在此基础上，导广义非线性模型度量影响的诊断统计量的二阶近似公式，作为推论，本文的结果适用于两种重要的特殊情形，第一，广义线性模型的影响分析，第二，正态非线性模型的影响分析。     

局部对称共形平坦空间的极小子流形（Ⅱ）

文［１］的重要结果推广到了环绕空间是局部对称共形平坦的情形，得到了这种空间中极小子流形截面曲率非负时，Ｒｉｃｃｉ曲率应满足的条件，做为应用，得到了比文［１］中结果更强的一个几何结果。