Regularized Riesz energies of submanifolds

Abstract

Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under Moebius transformations, thus giving a generalization to higher dimensions of the Moebius energy of knots.