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

TL;DR

This paper generalizes previous constructions of minimal surfaces in the three-sphere by allowing catenoidal bridges to concentrate along multiple parallel circles and poles, enabling dense distributions and potential applications to other minimal surface problems.

Contribution

It extends the Linearized Doubling methodology to construct minimal surfaces with multiple bridge concentrations, including dense distributions, and sets the stage for future applications.

Findings

Enables construction of minimal surfaces with densely distributed catenoidal bridges.

Shows potential limitations in creating embedded minimal surfaces with isolated singularities.

Provides a flexible framework for future doubling constructions in various geometries.

Abstract

In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ 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 before and the LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides (work in progress). In particular they allow us to develop in this forthcoming work 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. Our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be "densely distributed", that is do not miss any open set of $\mathbb{S}^2_{eq}$ in the limit. This in particular leads to interesting observations which 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.