Sharp Boundedness and Regularizing effects of the integral Menger   curvature for submanifolds
  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. 
Simon Blatt and S{\l}awomir Kolasi\'nski
math.FA math.MG