Doubling Constant Mean Curvature Tori in the 3-Sphere
Abstract
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.
Cite
@article{arxiv.math/0511742,
title = {Doubling Constant Mean Curvature Tori in the 3-Sphere},
author = {Adrian Butscher and Frank Pacard},
journal= {arXiv preprint arXiv:math/0511742},
year = {2007}
}
Comments
22 pages. Final version improves the statement of the theorem, correct some errors and improves the presentation. Accepted for publication by Annali SNS Pisa