Remark on a geometric inequality for closed hypersurfaces in weighted manifolds

In this paper we consider noncompact smooth metric measure spaces $(M,g,e^{-f}dvo{l}_g)$ of nonnegative Bakry-Émery Ricci curvature, i.e. $Ric+D^{2}f-\frac{1}{N}df\otimes df\ge 0$, for $0<N\le \mathrm{\infty }$, in order to obtain geometric inequalities for the boundary of a given open and bounded set $\mathrm{\Omega }\subset M$, with regular boundary $\mathrm{\partial }\mathrm{\Omega }$. Our inequalities are sharp for both the cases $N<\mathrm{\infty }$ and $N=\mathrm{\infty }$, provided that the underlying ambient space has large weighted volume growth. The rigidity obtained for the $N=\mathrm{\infty }$ case holds true precisely when $M\setminus \mathrm{\Omega }$ is isometric to a twisted product metric and, as such, is a generalization of the Willmore-type inequality for nonnegative Ricci curvature from Agostiniani, Fagagnolo and Mazzieri to the context of weighted manifolds.