Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds $(M,g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$\mathrm{.}WeproveasharpWillmore-typeinequalityforclosedhypersurfaces$\partial \Omega$in$M$,withequalityholdingtrueifandonlyif$(M{\setminus}\Omega, g)$isisometrictoatruncatedconeover$\partial\Omega$\mathrm{.}AnoptimalversionofHuiske{n}^{\mathrm{\prime }}sIsoperimetricInequalityfor$3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.