Volumetric Minkowski Inequality on Manifolds with Weighted Poincaré Inequality

Abstract In the paper, we establish a volumetric Minkowski inequality for complete manifolds admitting a weighted Poincaré inequality with the weight commensurable to the Ricci curvature lower bound. More precisely, we show that the weighted volume of a compact smooth domain in such manifolds is bounded from above by an integral of the mean curvature of its boundary. In particular, this implies that such manifolds admit no compact embedded minimal hypersurfaces. Examples of such manifolds abound and include both the manifolds with positive bottom spectrum and a large family of complete manifolds with nonnegative Ricci curvature, thus unifying the results by Munteanu and Wang in Munteanu, O., Wang, J.: A Minkowski type inequality for manifolds with positive spectrum. arXiv:2309.13749 (2023) and by Benatti, Fogagnolo and Mazzieri in Benatti, L., Fogagnolo, M., Mazzieri, L.: Minkowski inequality on complete Riemannian manifolds with nonnegative Ricci curvature. arXiv:2101.06063 (2021).