Doubling constant mean curvature tori in S3

Abstract

The Clifford tori in S 3 constitute a one-parameter family of flat, two- dimensional, constant mean curvature (CMC) submanifolds. This paper demon- strates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE meth- ods. That is, one can create a submanifold that has almost everywhere constant mean curvature by gluing a re-scaled catenoid into the neighbourhood of each point of a sub-lattice of the Clifford torus; and then one can show that a constant mean curvature perturbation of this submanifold does exist.