In this paper we show that embedded and compact C 1 manifolds have finite integral Menger curvature if and only if they are locally graphs of certain Sobolev-Slobodeckij spaces. Furthermore, we prove that for some intermediate energies of integral Menger type a similar characterization of objects with finite energy can be given.
In REF , Blatt and Kolasiński prove that a compact C 1 manifold has finite integral Menger curvatures (a higher dimensional version of Menger curvature) if and only if it can be locally represented by the graph of some Sobolev type map.
Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds
Mathematics