Willmore spacelike submanifolds in an indefinite space form $N_q^{n+p}\left(c\right)$

Shichang Shu, Junfeng Chen

Abstract: Let $N_q^{n+p}\left(c\right)$ be an $(n+p)$-dimensional connected indefinite space form of index $q$ $(1\le q\le p)$ and of constant curvature $c$. Denote by $\phi :M\to N_q^{n+p}\left(c\right)$ the $n$-dimensional spacelike submanifold in $N_q^{n+p}\left(c\right)$, $\phi :M\to N_q^{n+p}\left(c\right)$ is called a Willmore spacelike submanifold in $N_q^{n+p}\left(c\right)$ if it is a critical submanifold to the Willmore functional $W\left(\phi \right)={\int }_M{\rho }^{n}dv={\int }_M{(S-n{H}^2)}^{\frac{n}{2}}dv$, where $S$ and $H$ denote the norm square of the second fundamental form and the mean curvature of $M$ and ${\rho }^2=S-n{H}^2$. If $q=p$, in , we proved some integral inequalities of Simons’ type and rigidity theorems for $n$-dimensional Willmore spacelike submanifolds in a Lorentzian space form $N_p^{n+p}\left(c\right)$. In this paper, we continue to study this topic and prove some integral inequalities of Simons’ type and rigidity theorems for $n$-dimensional Willmore spacelike submanifolds in an indefinite space form $N_q^{n+p}\left(c\right)$ ($1\le q<p$).

Keywords: indefinite space form, Willmore spacelike submanifold, totally umbilical, Euler-Lagrange equation

Classification (MSC2000): 53C42; 53C40

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