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Doubling Constant Mean Curvature Tori in the 3-Sphere

Adrian Butscher·Frank Pacard
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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.

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Adrian Butscher, Frank Pacard. 2006-11-15. Doubling Constant Mean Curvature Tori in the 3-Sphere. https://arxiv.org/abs/math/0511742

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