Index of compact minimal submanifolds of the Berger spheres

Abstract

The stability and the index of compact minimal submanifolds of the Berger spheres S2n+1τ,0<τ≤1, are studied. Unlike the case of the standard sphere (τ=1), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if τ2≤1/2. Moreover, there are no stable compact minimal d-dimensional submanifolds of S2n+1τ when 1/(d+1)<τ2≤1 and the stable ones are classified for τ2=1/(d+1) when the submanifold is embedded. Finally, the compact orientable minimal surfaces of S3τ with index one are classified for 1/3≤τ2≤1.