摘要
考虑单位球面Sn+1(1)中的具有常平均曲率H的完备超曲面.在H≥0的假设下,通过计算两个式子知道,Clifford环面S1(a)×Sn-1(1-a2)对应的函数|Φ|是常数,并有两种可能性.通过深入研究这两种可能性,在球面的超曲面上定义的函数|Φ|,也具有de Sitter空间Sn1+1中常平均曲率H的完备类空超曲面相类似的现象,即有如下结论:对给定常数H≥0,记D±(H)=1/2(n/(n-1))^(1/2)[(n2H2+4(n-1))^(1/2)±(n-2)H].则有对任意的D∈[D-(H),D+(H)],都存在一个具有常平均曲率H的完备超曲面Mn→Sn+1,使得对应的函数|Φ|满足关系sup|Φ|=D.
In this paper,we consider complete hypersurfaces of the unit sphereSn+1(1).Under the assumption H≥0,through calculating two formulas,it can be known that the Clifford anchor ring S1(a)×Sn-1(1-a2) correspondence's function |Φ| is a constant,and has two possibilities.Through deep research these two possibilities,the function |Φ| which is defined in the spherical surface hypersurface,also has the phenomenon which is studied in paper[1] that a complete kind of spatial hypersurface with constant mean curvature H in the de Sitter space is similar.Namely the following conclusion is obtained.For each constant H≥0,we define two positive constants D±(H)=1/2(n/(n-1))^(1/2)[(n2H2+4(n-1))^(1/2)±(n-2)H],Then we have For any D∈[D-(H),D+(H)],there is a complete hypersurface Mn→Sn+1 with constant mean curvature H,such that the corresponding function |Φ| satisfies the relation sup|Φ|=D.
出处
《河南工程学院学报(自然科学版)》
2010年第4期61-63,共3页
Journal of Henan University of Engineering:Natural Science Edition
关键词
平均曲率
主曲率
第二基本形式
高斯方程
mean curvature
principal curvature
second fundamental form
gauss equations