Stable hypersurfaces with prescribed mean curvature

In recent works with N. Wickramasekera we develop a regularity and compactness theory for stable hypersurfaces (technically, integral varifolds) whose generalized mean curvature is prescribed by a (smooth enough) function on the ambient Riemannian manifold. I will describe the relevance of the theory to analytic and geometric problems, and describe some GMT and PDE aspects of the proofs.