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

Abstract

In the present paper we investigate generalizations of O'Hara's Möbius energy on curves \cite{ohara_1991a}, to Möbius-invariant energies on non-smooth subsets of ${\mathbb{R}}^n$ of arbitrary dimension and co-dimension. In particular, we show under mild assumptions on the local flatness of an admissible possibly unbounded set $\Sigma \subset {\mathbb{R}}^n$ that locally finite energy implies that $\Sigma$ is, in fact, an embedded Lipschitz submanifold of ${\mathbb{R}}^n$ -- 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 Möbius energy on closed curves. Our results hold in particular for Kusner and Sullivan's cosine energy $E_{\text{KS}}$ \cite{kusner-sullivan_1997} since one of the energies considered here is equivalent to $E_{\text{KS}}$.

39 pages, 1 figure, typos corrected, revised Cor. 3.12

References in corpus (2)

• Self-repulsiveness of energies for closed submanifolds
• A Möbius invariant discretization and decomposition of the Möbius energy