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Since every Riemannian manifold is locally Euclidean up to second order, small geodesic spheres of radius r have nearly constant mean curvature of magnitude 2/r. However, it is known that there are fairly restrictive conditions under which a geodesic sphere of sufficiently small radius can be perturbed to have exactly constant mean curvature. I will investigate a related question: whether it is possible to assemble small geodesic spheres into extended surfaces of near-constant mean curvature by the gluing technique, and perturbing these surfaces to have exactly constant mean curvature. It turns out that the conditions under which this is possible are the result of a subtle interplay between the surface and the background geometry. A result of my investigation is a construction of small constant mean curvature surfaces that locally resemble classical Delaunay surfaces but exhibit new global properties that are impossible in the classical setting. In addition, I will relate my investigation to the following question: given a sequence \Sigma_r of surfaces of mean curvature 2/r contained within a tubular neighbourhood of size O(r) of a lower-dimensional variety \Gamma in a Riemannian manifold M, what can be said about \Gamma?