Nonplanar minimal spheres in ellipsoids of revolution

TL;DR

This paper proves the existence of many nonplanar minimal 2-spheres in elongated ellipsoids of revolution, analyzing their growth and behavior as the ellipsoids approach a cylindrical shape.

Contribution

It introduces global bifurcation methods to demonstrate the existence and asymptotic properties of multiple minimal spheres in ellipsoids of revolution.

Findings

Existence of arbitrarily many nonplanar minimal spheres

Quantitative growth rate of minimal spheres

Asymptotic behavior as ellipsoids become cylindrical

Abstract

We use global bifurcation techniques to establish the existence of arbitrarily many geometrically distinct nonplanar embedded smooth minimal 2-spheres in sufficiently elongated 3-dimensional ellipsoids of revolution. More precisely, we quantify the growth rate of the number of such minimal spheres, and describe their asymptotic behavior as the ellipsoids converge to a cylinder.