Doubling Constant Mean Curvature Tori in the 3-Sphere
Abstract
Description
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.
22 pages. Final version improves the statement of the theorem, correct some errors and improves the presentation. Accepted for publication by Annali SNS Pisa
22 pages. Final version improves the statement of the theorem, correct some errors and improves the presentation. Accepted for publication by Annali SNS Pisa