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

In this paper, we introduce a special class of hypersurfaces which are called λ -hypersurfaces related to a weighted volume preserving mean curvature flow in the 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 ≥ 0 . The classification result generalizes the results of Huisken (J Differ Geom 31:285–299, 1990 ) and Colding and Minicozzi (Ann Math 175:755–833, 2012 ).