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

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

2008-04-23

arxiv.org/abs/0804.3920v1

Tokyo J. Math. 31 (2008), 161-174

Mathematics

Differential Geometry

This is a longer version of a paper to appear in Tokyo J. Math., with extra numerical data included

Scientific paper

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.

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