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

Shu, Shichang. "Hypersurfaces with constant $k$-th mean curvature in a Lorentzian space form." Archivum Mathematicum 046.2 (2010): 87-97. <http://eudml.org/doc/37769>.

@article{Shu2010,
abstract = {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 $.},
author = {Shu, Shichang},
journal = {Archivum Mathematicum},
keywords = {space-like hypersurface; Lorentzian space form; $k$-mean curvature; principal curvature; space-like hypersurface; Lorentzian space form; -mean curvature; principal curvature},
language = {eng},
number = {2},
pages = {87-97},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Hypersurfaces with constant $k$-th mean curvature in a Lorentzian space form},
url = {http://eudml.org/doc/37769},
volume = {046},
year = {2010},
}

TY - JOUR
AU - Shu, Shichang
TI - Hypersurfaces with constant $k$-th mean curvature in a Lorentzian space form
JO - Archivum Mathematicum
PY - 2010
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 046
IS - 2
SP - 87
EP - 97
AB - 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 $.
LA - eng
KW - space-like hypersurface; Lorentzian space form; $k$-mean curvature; principal curvature; space-like hypersurface; Lorentzian space form; -mean curvature; principal curvature
UR - http://eudml.org/doc/37769
ER -