Volume estimates and classification theorem for constant weighted mean curvature hypersurfaces

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $\mathrm{\Sigma }\subset {\mathbb{R}}^{n+1}$. 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.