Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere by Gluing Spherical Building Blocks

TL;DR

This paper extends gluing techniques for constructing constant mean curvature hypersurfaces in the (n+1)-sphere to less symmetric configurations, identifying necessary geometric conditions for exact solutions.

Contribution

It generalizes previous methods to handle less symmetric initial configurations and establishes new global geometric balancing conditions.

Findings

Gluing techniques are extended to less symmetric configurations.

Balancing conditions are necessary for perturbing to exact CMC hypersurfaces.

Examples demonstrate configurations with and without symmetries.

Abstract

The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere 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 (n+1)-sphere 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.