Bounding volume of compact domains via mean curvature and bottom spectrum

Abstract The classical Minkowski inequality implies that the volume of a bounded convex domain in the Euclidean space is controlled from above by the integral of the mean curvature of its boundary. In this note, an analogous inequality is established without assuming convexity, valid for all bounded smooth domains in a complete manifold whose bottom spectrum is suitably large relative to its Ricci curvature lower bound. An immediate consequence is the nonexistence of embedded closed minimal hypersurfaces in such manifolds. The same nonexistence issue is also addressed for steady and expanding Ricci solitons. The proofs are very much inspired by a sharp monotonicity formula, derived for positive harmonic functions on manifolds with positive spectrum.