Willmore spacelike submanifolds in a Lorentzian space form $N^{n+p}_p(c)$

Abstract

Let $N^{n+p}_p(c)$ be an $(n+p)$-dimensional connected Lorentzian space form of constant sectional curvature $c$ and $\varphi: M \rightarrow N^{n+p}_p(c)$ an $n$-dimensional spacelike submanifold in $N^{n+p}_p(c)$. The immersion $\varphi: M \rightarrow N^{n+p}_p(c)$ is called a Willmore spacelike submanifold in $N^{n+p}_p(c)$ if it is a critical submanifold to the Willmore functional
\[
W(\varphi)=\int_M\rho^ndv=\int_M(S-nH^2)^{\frac{n}{2}}dv,
\]
where $S$, $H$ and $\rho^2$ denote the norm square of the second fundamental form, the mean curvature and the non-negative function
$\rho^2=S-nH^2$ of $M$. In this article, by calculating the first variation of $W(\varphi)$, we obtain the Euler-Lagrange equation of $W(\varphi)$ and prove some rigidity theorems for $n$-dimensional Willmore spacelike submanifolds in $N^{n+p}_p(c)$.