Instability of Hopf vector fields on Lorentzian Berger spheres

TL;DR

This paper investigates the stability of Hopf vector fields on Lorentzian Berger spheres, demonstrating their instability through spectral analysis and extending the approach to open problems in the Riemannian setting.

Contribution

It introduces a spectral method using eigenfunctions to analyze the stability of Hopf vector fields on Lorentzian Berger spheres and applies this to unresolved issues in Riemannian geometry.

Findings

Hopf vector fields are unstable as critical points of energy, volume, and generalized energy.

Constructed eigenfunction-based vector fields to analyze Hessians of functionals.

Extended the spectral technique to address open problems in Riemannian geometry.

Abstract

In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.