| Title:
|
Complete spacelike hypersurfaces with constant scalar curvature (English) |
| Author:
|
Shu, Schi Chang |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
105-114 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we characterize the $n$-dimensional $(n\ge 3)$ complete spacelike hypersurfaces $M^n$ in a de Sitter space $S^{n+1}_1$ with constant scalar curvature and with two distinct principal curvatures one of which is simple.We show that $M^n$ is a locus of moving $(n-1)$-dimensional submanifold $M^{n-1}_1(s)$, along $M^{n-1}_1(s)$ the principal curvature $\lambda $ of multiplicity $n-1$ is constant and $M^{n-1}_1(s)$ is umbilical in $S^{n+1}_1$ and is contained in an $(n-1)$-dimensional sphere $S^{n-1}\big (c(s)\big )=E^n(s)\cap S^{n+1}_1$ and is of constant curvature $\big (\frac{d\lbrace \log |\lambda ^2-(1-R)|^{1/n}\rbrace }{ds}\big )^2-\lambda ^2+1$,where $s$ is the arc length of an orthogonal trajectory of the family $M^{n-1}_1(s)$, $E^n(s)$ is an $n$-dimensional linear subspace in $R^{n+2}_1$ which is parallel to a fixed $E^n(s_0)$ and $u=\big |\lambda ^2-(1-R)\big |^{-\frac{1}{n}}$ satisfies the ordinary differental equation of order 2, $\frac{d^2u}{ds^2}-u\big (\pm \frac{n-2}{2}\frac{1}{u^n}+R-2\big )=0$. (English) |
| Keyword:
|
de Sitter space |
| Keyword:
|
spacelike hypersurface |
| Keyword:
|
scalar curvature |
| Keyword:
|
principal curvature |
| Keyword:
|
umbilical |
| MSC:
|
53C20 |
| MSC:
|
53C42 |
| idZBL:
|
Zbl 1212.53084 |
| idMR:
|
MR2432847 |
| . |
| Date available:
|
2008-07-24T13:17:45Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/116927 |
| . |
| Reference:
|
[1] Brasil, A., Jr., , Colares, A. G., Palmas, O.: Complete spacelike hypersurfaces with constant mean curvature in the de Sitter space: A gap Theorem.Illinois J. Math. 47 (3) (2003), 847–866. Zbl 1047.53031, MR 2007240 |
| Reference:
|
[2] Cheng, Q. M.: Complete hypersurfaces in a Euclidean space $R^{n+1}$ with constant scalar curvature.Indiana Univ. Math. J. 51 (2002), 53–68. MR 1896156, 10.1512/iumj.2002.51.2040 |
| Reference:
|
[3] Otsuki, T.: Minimal hypersurfaces in a Riemannian manifold of constant curvature.Amer. J. Math. 92 (1970), 145–173. Zbl 0196.25102, MR 0264565, 10.2307/2373502 |
| Reference:
|
[4] Shu, S. C.: Complete spacelike hypersurfaces in a de Sitter space.Bull. Austral. Math. Soc. 73 (2006), 9–16. Zbl 1098.53051, MR 2206558, 10.1017/S0004972700038570 |
| Reference:
|
[5] Zheng, Y.: On spacelike hypersurfaces in the de Sitter spaces.Ann. Global Anal. Geom. 13 (1995), 317–321. MR 1364006, 10.1007/BF00773403 |
| Reference:
|
[6] Zheng, Y.: Spacelike hypersurfaces with constant scalar curvature in the de Sitter spaces.Differential Geom. Appl. 6 (1996), 51–54. MR 1384878, 10.1016/0926-2245(96)00006-X |
| . |
