Instability of Hopf vector fields on Lorentzian Berger spheres

Details Instability of Hopf vector fields on Lorentzian Berger spheres Instability of Hopf vector fields on Lorentzian Berger spheres

2008-03-17

arxiv.org/abs/0803.2487v1

Mathematics

Differential Geometry

18 pages, 1 figure

Scientific paper

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.

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