S^(n+1)(1)中满足D_H^-≤sup|Φ|≤D_H^+的完备超曲面

Complete Hypersurfaces Satisfying D_H^-≤sup|Φ|≤D_H^+ in S^(n+1)(1)

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.