| Title:
|
Hypersurfaces with constant $k$-th mean curvature in a Lorentzian space form (English) |
| Author:
|
Shu, Shichang |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
46 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
87-97 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we study $n(n\ge 3)$-dimensional complete connected and oriented space-like hypersurfaces $M^n$ in an (n+1)-dimensional Lorentzian space form $M^{n+1}_1(c)$ with non-zero constant $k$-th $(k<n)$ mean curvature and two distinct principal curvatures $\lambda $ and $\mu $. We give some characterizations of Riemannian product $H^m(c_1)\times M^{n-m}(c_2)$ and show that the Riemannian product $H^m(c_1)\times M^{n-m}(c_2)$ is the only complete connected and oriented space-like hypersurface in $M^{n+1}_1(c)$ with constant $k$-th mean curvature and two distinct principal curvatures, if the multiplicities of both principal curvatures are greater than 1, or if the multiplicity of $\lambda $ is $n-1$, $\lim \limits _{s\rightarrow \pm \infty }\lambda ^k\ne H_k$ and the sectional curvature of $M^n$ is non-negative (or non-positive) when $c>0$, non-positive when $c\le 0$, where $M^{n-m}(c_2)$ denotes $R^{n-m}$, $S^{n-m}(c_2)$ or $H^{n-m}(c_2)$, according to $c=0$, $c>0$ or $c<0$, where $s$ is the arc length of the integral curve of the principal vector field corresponding to the principal curvature $\mu $. (English) |
| Keyword:
|
space-like hypersurface |
| Keyword:
|
Lorentzian space form |
| Keyword:
|
$k$-mean curvature |
| Keyword:
|
principal curvature |
| MSC:
|
53A10 |
| MSC:
|
53C42 |
| idZBL:
|
Zbl 1240.53101 |
| idMR:
|
MR2684251 |
| . |
| Date available:
|
2010-06-22T22:11:07Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140305 |
| . |
| Reference:
|
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