Complete $\lambda$-hypersurfaces of weighted volume-preserving mean curvature flow

In this paper, we introduce a definition of $\lambda$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $\lambda$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $\lambda$-hypersurfaces with polynomial area growth and $H-\lambda\geq 0$, which are generalizations of the results due to Huisken, Colding-Minicozzi. We also define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $\lambda$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of the area for complete and non-compact $\lambda$-hypersurfaces are also studied.