Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Abstract

We study a modified version of Lerman-Whitehouse Menger-like curvature defined for m+2 points in an n-dimensional Euclidean space. For 1 <= l <= m+2 and an m-dimensional subset S of R^n we also introduce global versions of this discrete curvature, by taking supremum with respect to m+2-l points on S. We then define geometric curvature energies by integrating one of the global Menger-like curvatures, raised to a certain power p, over all l-tuples of points on S. Next, we prove that if S is compact and m-Ahlfors regular and if p is greater than ml, then the P. Jones' β-numbers of S must decay as r^t with r \to 0 for some t in (0,1). If S is an immersed C^1 manifold or a bilipschitz image of such set then it follows that it is Reifenberg flat with vanishing constant, hence (by a theorem of David, Kenig and Toro) an embedded C^{1,t} manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent t to the optimal one a = 1 - ml/p showing an analogue of the Morrey-Sobolev embedding theorem. Moreover, we obtain a qualitative control over the local graph representations of S only in terms of the energy.

I removed Example 3.11, which was wrong in the sense that the β-numbers for this set do not decay as r^2

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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
• Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies
• Self-repulsiveness of energies for closed submanifolds
• Integral Menger Curvature and Rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space
• 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