  The Clifford tori in the 3-sphere are a one-parameter family of flat,
two-dimensional, constant mean curvature (CMC) surfaces. This paper
demonstrates 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
methods. That is, one can create an approximate solution by gluing a rescaled
catenoid into the neighbourhood of each sub-lattice point; and then one can
show that a perturbation of this approximate submanifold exists which satisfies
the CMC condition.
