Abstract: We are interested in families of constant mean curvature hypersurfaces, with mean curvature varying from one member of the family to another, which `condense' to a submanifold $K^k \subset M^{m+1}$ of codimension greater than $1$.

Two cases have been studied previously: R. Ye proved the existence of a local foliation by constant mean curvature hypersurfaces when $K$ is a point (which is required to be a nondegenerate critical point of the scalar curvature function) and R. Mazzeo and F. Pacard proved existence of a lamination when $K$ is a nondegenerate geodesic.

We extend the result of Mazzeo and Pacard to handle the general case, when $K$ is an arbitrary nondegenerate minimal submanifold. In particular, this proves the existence of constant mean curvature hypersurfaces with nontrivial topology in any Riemannian manifold. We use a new approach inspired by some recent work of A. Malchiodi and M. Montenegro in the contex of semilinear elliptic partial differential equations. (work with R. Mazzeo and F.Pacard)