  In this paper, we study $\lambda$-submanifolds of arbitrary codimensions in
Gauss spaces. These submanifolds can be seen as natural generalizations of
self-shrinker and $\lambda$-hypersurfaces. Using a divergence type theorem and
some Simons' type identities, we prove some halfspace type theorems and gap
theorems for complete proper $\lambda$-submanifolds. These generalized our as
well as the others' results for self-shrinker or $\lambda$-hypersurfaces to
$\lambda$-submanifolds.
