CMC tori of revolution in $\mathbb{S}^3$: additional data on the spectra of their Jacobi operators

TL;DR

This paper develops numerical methods to analyze the spectra of elliptic operators on closed loops and applies them to compute the Morse index of constant mean curvature tori of revolution in the 3-sphere, confirming a lower bound.

Contribution

It introduces accurate numerical techniques for spectral analysis of elliptic operators on loops and applies these to determine the Morse index of specific geometric surfaces.

Findings

Every constant mean curvature torus of revolution in S^3 has Morse index at least five.

The numerical methods confirm known lower bounds are close to optimal.

The work provides additional spectral data for these tori.

Abstract

We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit 3-sphere $\mathbb{S}^3$, confirming that every such torus has Morse index at least five, and showing that other known lower bounds for this Morse index are close to optimal.