TY - JOUR
ID - 20589
T1 - $L_k$-biharmonic spacelike hypersurfaces in Minkowski $4$-space $mathbb{E}_1^4$
JO - Sahand Communications in Mathematical Analysis
JA - SCMA
LA - en
SN - 2322-5807
A1 - Pashaie, Firooz
A1 - Mohammadpouri, Akram
Y1 - 2017
PY - 2017/01/01
VL - 05
IS - 1
SP - 21
EP - 30
KW - Spacelike hypersurface
KW - Biharmonic
KW - $L_k$-biharmonic
KW - $k$-maximal
DO - 10.22130/scma.2017.20589
N2 - Biharmonic surfaces in Euclidean space $mathbb{E}^3$ are firstly studied from a differential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. A surface $x : M^2rightarrowmathbb{E}^{3}$ is called biharmonic if $Delta^2x=0$, where $Delta$ is the Laplace operator of $M^2$. We study the $L_k$-biharmonic spacelike hypersurfaces in the $4$-dimentional pseudo-Euclidean space $mathbb{E}_1^4$ with an additional condition that the principal curvatures are distinct. A hypersurface $x: M^3rightarrowmathbb{E}^{4}$ is called $L_k$-biharmonic if $L_k^2x=0$ (for $k=0,1,2$), where $L_k$ is the linearized operator associated to the first variation of $(k+1)$-th mean curvature of $M^3$. Since $L_0=Delta$, the matter of $L_k$-biharmonicity is a natural generalization of biharmonicity. On any $L_k$-biharmonic spacelike hypersurfaces in $mathbb{E}_1^4$ with distinct principal curvatures, by, assuming $H_k$ to be constant, we get that $H_{k+1}$ is constant. Furthermore, we show that $L_k$-biharmonic spacelike hypersurfaces in $mathbb{E}_1^4$ with constant $H_k$ are $k$-maximal.
UR - http://scma.maragheh.ac.ir/article_20589.html
L1 - http://scma.maragheh.ac.ir/pdf_20589_41cae243cd77692b496d7ab7a304e79b.html
ER -