Embedded constant mean curvature hypersurfaces

Reserach Areas:
Primary research area: Geometry and Topology.
Secondary research area: Mathematical Analysis

Recent has shown that in any compact Riemannian manifold of dimension 3 or higher, and given any non-zero number a, there exists a closed constant mean curvature hypersurface with scalar mean curvature equal to a. The hypersurface may have a singular set of codimension 7 and is quasi-embedded, namely it is the image of an immersion and fails to be an embedding only at points around which the structure is given by the union of exactly two embedded disks that lie on one side of each other and intersect tangentially.
The scope of this project is to investigate assumptions on the Riemannian metric under which it is possible to find a constant mean curvature hypersurface as above that is additionally embedded (rather than quasi-embedded) away from a singular set of codimension 7. This may be true for metrics with positive Ricci curvature and, in low dimensions, for bumpy metrics (a generic set of metrics for which the second variation of smooth minimal hypersurfaces has trivial nullity).