Stability of Unit Hopf Vector Fields on Quotients of Spheres

Vincent Borrelli (1) , Hanifi Zoubir (1)

1 ICJ - Institut Camille Jordan

The volume of a unit vector field $V$ of a Riemannian manifold $(M,g)$ ($n$ odd) is the volume of its image $V(M^n)$ in the unit tangent bundle endowed with the Sasaki metric. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fiber of a Hopf fibration $\Sph^n\ra \C P^{\frac{n-1}{2}}$ are well known to be critical for the volume functional on the round $n$-dimensional sphere $\Sph^n(r)$ for every radius $r>1$. Regarding the Hessian, it turns out that its positivity actually depends on the radius. Indeed, in \cite{B-GM}, it is proven that for $n\geq 5$ there is a critical radius $r_c=\frac{1}{\sqrt{n-4}}$ such that Hopf vector fields are stable if and only if $r\leq r_c.$ In this paper we consider the question of the existence of a critical radius for space forms $M^n(c)$ ($n$ odd) of positive curvature $c$. These space forms are isometric quotients $\Sph^n(r)/\Gamma$ of round spheres and naturally carry a unit Hopf vector field which is critical for the volume functional. We prove that $r_c=+\infty$, unless $\Gamma$ is trivial. So, in contrast with the situation for the sphere, the Hopf field is stable on $\Sph^n(r)/\Gamma$, $\Gamma\neq\{Id\}$, whatever the radius.

Mots clés: Volume; Hopf vector field; Stability; Hessian; Space form

Domaines: Géométrie différentielle [math.DG]

Vincent Borrelli, Hanifi Zoubir. Stability of Unit Hopf Vector Fields on Quotients of Spheres. 2009. ⟨hal-00363399⟩