Minimal Surfaces in the Round Three‐Sphere by Doubling the Equatorial Two‐Sphere, II
Nikolaos Kapouleas, Peter McGrath · Communications on Pure and Applied Mathematics · 2019
Abstract In a previous paper, new closed embedded smooth minimal surfaces in the round three‐sphere were constructed, each resembling two parallel copies of the equatorial two‐sphere 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. Unlike in the previous paper, our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be “densely distributed,” that is, they do not miss any open set of in the limit. This in particular leads to interesting observations that seem to suggest that it may be impossible to construct embedded minimal surfaces with isolated singularities by concentrating infinitely many catenoidal necks at a point. © 2019 Wiley Periodicals, Inc.