Isoperimetric inequalities in cylinders with density

Abstract

Given a compact Riemannian manifold with density M without boundary and the real line R with constant density, we prove that isoperimetric regions of large volume in M×R with the product density are slabs of the form M×[a,b]. We previously prove, as a necessary step, the existence of isoperimetric regions in any manifold of density where a subgroup of the group of transformations preserving weighted perimeter and volume acts cocompactly.