Variational characterizations of $ξ$-submanifolds in the Eulicdean space $\bbr^{m+p}$

$ξ$-submanifold in the Euclidean space $\bbr^{m+p}$ is a natural extension of the concept of self-shrinker to the mean curvature flow in $\bbr^{m+p}$. It is also a generalization of the $λ$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $ξ$-submanifolds are established. First, it is shown that a submanifold in $\bbr^{m+p}$ is a $ξ$-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space $(\bbr^{m+p},e^{-\fr{|x|^2}{m}}\lagl\cdot,\cdot\ragl)$; Then, two weighted volume functionals $V_ξ$ and $\bar V_ξ$ are introduced and it is proved that $ξ$-submanifolds can be characterized as the critical points of these two functionals; Also, the corresponding second variation formulas are computed and the ($W$-)stability properties for $ξ$-submanifolds are systematically studied. In particular, it is proved that $m$-planes are the only properly immersed, complete $W$-stable $ξ$-submanifolds with flat normal bundle under a technical condition. It would be interesting if this additional restriction could be removed.

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