Constant mean curvature hypersurfaces in ${{\mathbb S}^{n+1}}$ by gluing spherical building blocks

Abstract

The techniques developed by Butscher (Gluing constructions amongst constant mean curvature hypersurfaces of ${{\mathbb S}^{n+1}}$) for constructing constant mean curvature (CMC) hypersurfaces in ${{\mathbb S}^{n+1}}$ by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These balancing conditions are analogous to those that must be satisfied in the “classical” context of gluing constructions of CMC hypersurfaces in Euclidean space, although they are more restrictive in the ${{\mathbb S}^{n+1}}$ case. An example of an initial configuration is given which demonstrates this fact; and another example of an initial configuration is given which possesses no symmetries at all.