Stability of unit Hopf vector fields on quotients of spheres (Q980905)

The goal of this work is to investigate the problem of the existence of a critical radius for quotients of spheres. Any space form with positive sectional curvature is a quotient of \(S^{2m-1}(r)\) by a finite fixed point free isometry subgroup \(\Gamma\) of \(O(2m+2)\). It turns out that there is still (at last) a Hopf field \(H_r\) on the quotient \(S^{2m+1}(r)/\Gamma\) which remains critical for volume functional. But something a priori unexpected occurs: the field \(H_r\) is always stable, whatever the radius \(r> 0\). Theorem. Let \(M= S^{2m+1}(r)/\Gamma\) be a space form with \(\Gamma\neq(\text{id})\), then the Hopf field \(H_r\) is stable on \(S^{2m+1}(r)/\Gamma\).