Complete $\lambda$-submanifolds in Gauss spaces

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.