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

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