Frank Morgan

An ingredient in proving the existence of isoperimetric regions and clusters of prescribed volume(s) is the boundedness of isoperimetric regions of smaller volume(s). One proof of boundedness is by monotonicity. The proof in my Geometric Measure Theory book (Lemma 13.6) uses a non-sharp isoperimetric inequality for small volume and has the advantage of applying to convex integrands more general than area; the requisite isoperimetric inequality follows immediately from the isoperimetric inequality for area. All of this works equally well in the presence of a density. For more delicate results see

Frank Morgan and Aldo Pratelli, Existence of isoperimetric regions in Rⁿ with density, Ann. Global Anal. Geom. (2012); arXiv.org (2011)

and Cinti and Pratelli.

My book comments that existence similarly holds in any smooth Riemannian manifold M with compact quotient M/G by a group of isometries. Monotonicity still yields boundedness; even with density (bounded above and below by compactness) the classical mean curvature is bounded and hence classical monotonicity applies. To use my alternate proof, which applies to more general integrands, one needs the isoperimetric inequality for small volume, which follows immediately from such an isoperimetric inequality for area without density. For the latter in a more general setting, see Theorem 2.1 of Morgan and Ritoré “Isoperimetric regions in cones” after Berard and Meyer. In this simpler setting of M/G compact, you can just cover M with congruent balls with a bounded number of balls meeting each point and apply the relative isoperimetric inequality in each ball. Incidentally the same covering can be used to replace the division into cubes in the existence proof 13.7 in my book.