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

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 $E_\textnormal{KS}$.