Stability and index of compact minimal submanifolds

A submanifold of a Riemannian manifold is called minimal if its mean curvature vector vanish everywhere. These manifolds appear as critical points of the volume functional. From this point of view, we will introduce the second variation of the volume and study those compact minimal submanifolds with positive semi-definite second variation, that are called stable. We will show that the sphere does not admit compact stable minimal submanifolds and we will characterize those compact minimal submanifolds which fail short from being stable (the lowest index examples). If time allows, we will also discuss some generalizations of these results to the Berger spheres (aka spheres endowed with a deformation of the round metric in the direction of the Hopf fibers).