Generalized doubling constructions for constant mean curvature hypersurfaces in Sn+1

The sphere Sn+1 contains a simple family of constant mean curvature (CMC) hypersurfaces of the form Ct := Sp(cos t) × Sq(sin t) for p + q = n and t ∈ (0, π2 ) called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each point; and then one can show that a perturbation of this approximate hypersurface exists, which satisfies the CMC condition. The results of this paper generalize those of the authors in [3].