Regularized Riesz energies of submanifolds

TL;DR

This paper investigates regularized Riesz energies of submanifolds, comparing regularization methods and establishing invariance under Moebius transformations, extending knot energy concepts to higher dimensions.

Contribution

It introduces a comparison of regularization techniques for divergent Riesz energies and demonstrates Moebius invariance, generalizing knot energy concepts to higher-dimensional submanifolds.

Findings

Hadamard and analytic continuation regularizations agree for Riesz energies.

Certain Riesz energies are invariant under Moebius transformations.

Extension of knot energy invariance to higher-dimensional 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.