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

Abstract

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.

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Cited by in corpus (7)

• On some knot energies involving Menger curvature
• Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures
• Geometric Sobolev-like embedding using high-dimensional Menger-like curvature
• Minimal Hölder regularity implying finiteness of integral Menger curvature
• Self-repulsiveness of energies for closed submanifolds
• On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case
• M{ö}bius-invariant self-avoidance energies for non-smooth sets in arbitrary dimensions