On the second variation of the Graham-Witten energy

The area renormalization procedure gives an invariant of even-dimensional closed submanifolds in a conformal manifold, which we call the Graham-Witten energy, and it is a generalization of the classical Willmore energy. In this paper, we obtain an explicit formula for the second variation of this energy at minimal submanifolds in an Einstein manifold. As an application, we prove that the even-dimensional totally geodesic spheres in the unit sphere are critical points of the Graham-Witten energy with non-negative second variation.