Index and nullity of a family of harmonic tori in the sphere

TL;DR

This paper studies the stability and degeneracy properties of a specific family of harmonic tori in the sphere by analyzing their index and nullity, providing insights into their geometric and analytical characteristics.

Contribution

It introduces a detailed analysis of the index and nullity for vacuum solutions from the two-torus into the sphere, a novel focus in harmonic map theory.

Findings

Computed the index and nullity for the family of harmonic tori.

Identified conditions under which the solutions are stable or degenerate.

Enhanced understanding of the spectral properties of harmonic tori in the sphere.

Abstract

In this note we investigate the $(E)$-index and the $(E)$-nullity of vacuum solutions from the two-torus into the two sphere.