Minimal Surfaces in the Round Three‐Sphere by Doubling the Equatorial Two‐Sphere, II

By N. Kapouleas, Peter J. McGrath. Communications on Pure and Applied Mathematics, 2014

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In a previous paper, new closed embedded smooth minimal surfaces in the round three‐sphere S31 were constructed, each resembling two parallel copies of the equatorial two‐sphere Seq2 joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, or the equatorial circle and the poles. In this sequel, we generalize those constructions so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, with the further option to include bridges at the poles. The current constructions follow the linearized doubling (LD) methodology developed in the previous paper. The LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides [15]. In particular, they allow us to develop doubling constructions for the catenoid in Euclidean three‐space, the critical catenoid in the unit ball, and the spherical shrinker of the mean curvature flow.

Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/20717495eac3be044a60d6131402d1432319aa70