Volume and energy stability for isometric minimal immersions (Q1113108)

It is well known that a minimal isometric immersion from a Riemannian manifold M into a Riemannian manifold N is also a harmonic immersion. The second order behavior of the volume and energy functionals is studied in this paper, and it is shown that in the case of codimension-one immersions into flat Riemannian manifolds, the immersions are always energy stable, but will be volume stable if and only if they are totally geodesic. Several examples of energy stable, but volume unstable minimal surfaces in the three-torus are then described. Similar results are given for minimal isometric immersions into manifolds of codimension greater than one, provided the target manifold is parallelizable.