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.

We propose a notion of integral Menger curvature for compact, m-dimensional sets in n-dimensional Euclidean space and prove that finiteness of this quantity implies that the set is C 1,α embedded manifold with the Hölder norm and the size of maps depending only on the curvature. We develop the ideas introduced by Strzelecki and von der Mosel [Adv. Math. 226(2011)] and use a similar strategy to prove our results.

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Abstract. We study two kinds of integral Menger-type curvatures. We find a threshold value of α 0 , a Hölder exponent, such that for all α > α 0 embedded C 1,α manifolds have finite curvature. We also give an example of a C 1,α 0 injective curve and higher dimensional embedded manifolds with unbounded curvature.