In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces Σ ⊂ R n+1 . We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume and exponential volume growth for submanifolds with weighted mean curvature of at most linear growth.Example 1.1. Any self-shrinker is a CWMC hypersurface with λ = 0.Example 1.2. All hyperplanes in R n+1 are CWMC hypersurfaces with λ = ± d 2 , where d denotes the distance from the hyperplane to the origin and the sign depends on the orientation. Indeed, let Σ ⊂ R n+1 be a hyperplane and x 0 ∈ Σ such that d(Σ, 0) = d(x 0 , 0) = d. This implies that ±d = x 0 , ν , Key words and phrases. weighted mean curvature, exponential volume growth, finite weighted volume. This study was financed in part by the Coordenao de Aperfeioamento de Pessoal de Nvel Superior -Brasil (CAPES) Finance Code 001 .

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