M{\"o}bius-invariant self-avoidance energies for non-smooth sets in arbitrary dimensions

TL;DR

This paper extends M"obius-invariant energies to non-smooth sets in arbitrary dimensions, showing finite energy implies certain regularity and vice versa, with implications for understanding geometric properties of sets.

Contribution

It introduces a generalization of M"obius energies to higher-dimensional, non-smooth sets and establishes equivalences between finite energy and regularity properties.

Findings

Finite energy implies the set is an embedded Lipschitz submanifold.

Low fractional Sobolev regularity ensures finite energy.

Results apply to Kusner and Sullivan's cosine energy, showing its equivalence.

Abstract

In the present paper we investigate generalizations of O'Hara's M\"obius energy on curves \cite{ohara_1991a}, to M\"obius-invariant energies on non-smooth subsets of $\R^n$ of arbitrary dimension and co-dimension. In particular, we show under mild assumptions on the local flatness of an admissible possibly unbounded set $\Sigma\subset \R^n$ that locally finite energy implies that $\Sigma$ is, in fact, an embedded Lipschitz submanifold of $\R^n$ -- sometimes even smoother (depending on the a priorily given additional regularity of the admissible set). We also prove, on the other hand, that a local graph structure of low fractional Sobolev regularity on a set $\Sigma$ is already sufficient to guarantee finite energy of $\Sigma$. This type of Sobolev regularity is exactly what one would expect in view of Blatt's characterization \cite{blatt_2012a} of the correct energy space for the M\"obius energy on closed curves. Our results hold in particular for Kusner and Sullivan's cosine energy $E_\textnormal{KS}$ \cite{kusner-sullivan_1997} since one of the energies considered here is equivalent to $E_\textnormal{KS}$.