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In this paper we show that embedded and compact $C^1$ manifolds have finiteintegral Menger curvature if and only if they are locally graphs of certainSobolev-Slobodeckij spaces. Furthermore, we prove that for some intermediateenergies of integral Menger type a similar characterization of objects withfinite energy can be given.

Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds