Bifurcations of Clifford tori in ellipsoids

TL;DR

This paper proves that most 3D ellipsoids with a 2-torus symmetry contain infinitely many minimal tori, which bifurcate from the largest volume orbit and are invariant under a circle, revealing rich geometric structures.

Contribution

It establishes the existence of infinitely many minimal tori in ellipsoids with 2-torus symmetry, detailing their bifurcation from specific orbits at dense eccentricities.

Findings

Infinitely many minimal tori exist in most ellipsoids with 2-torus symmetry.

Minimal tori bifurcate from the largest volume orbit at dense eccentricities.

Most minimal tori remain invariant under a circle.

Abstract

We prove that 3-dimensional ellipsoids invariant under a 2-torus action contain infinitely many distinct immersed minimal tori, with at most one exception. These minimal tori bifurcate from the 2-torus orbit of largest volume at a dense set of eccentricities, and remain invariant under a circle.