Complete $λ$-hypersurfaces of weighted volume-preserving mean curvature flow

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

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