Hypersurfaces with constant $k$-th mean curvature in a Lorentzian space form

Shu, Shichang

Archivum mathematicum, Tome 46 (2010) no. 2, pp. 87-97.

Résumé

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 $(k$ 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 $c0$, where $s$ is the arc length of the integral curve of the principal vector field corresponding to the principal curvature $\mu $.

Détail

Classification : 53A10, 53C42
Keywords: space-like hypersurface; Lorentzian space form; $k$-mean curvature; principal curvature