On the geometry of CMC hyper surfaces embedded in a manifold of dimension 4 or 5

Properties of constant mean curvature surfaces, with assumptions on completeness, compactness, bounded second fundamental form and connectedness. Currently, I am proving that a simply connected, complete, constant mean curvature surface with bounded second fundamental form is either compact or not contained in an extrinsic ball. I am using techniques from my supervisor's paper and from similar results that are for minimal surfaces only.

We are investigating CMC hypersurfaces embedded in Euclidean and non-Euclidean space which have bounded second fundamental form, are non-compact and are complete.