Möbius-invariant self-avoidance energies for non-smooth sets of arbitrary dimension and co-dimension

In the present paper we investigate generalizations of O'Hara's Möbius energy on curves [40], to Möbius-invariant energies on non-smooth subsets of Rn of arbitrary dimension and co-dimension. In particular, we show under mild assumptions on the local flatness of an admissible possibly unbounded set Σ⊂Rn that locally finite energy implies that Σ is, in fact, an embedded Lipschitz submanifold of Rn – sometimes even smoother (depending on additional regularity assumptions on the admissible set). We also prove, on the other hand, that a local graph structure of low fractional Sobolev regularity on a set Σ is already sufficient to guarantee finite energy of Σ. This type of Sobolev regularity is exactly what one would expect in view of Blatt's characterization [5] of the correct energy space for the Möbius energy on closed curves. Our results hold in particular for Kusner and Sullivan's cosine energy EKS[37] since one of the energies considered here is equivalent to EKS.