Integral Menger curvature for surfaces

Abstract

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda }$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale $R>0$ which depends only on an upper bound $E$ for the integral Menger curvature $M_p(\Sigma )$ and the integrability exponent $p$, and \emph{not} on the surface $\Sigma$ itself; below that scale, each surface with energy smaller than $E$ looks like a nearly flat disc with the amount of bending controlled by the (local) $M_p$-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent $\lambda$ up to the optimal one, $\lambda =1-(8/p)$, thus establishing a new geometric `Morrey-Sobolev' imbedding theorem. As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.

64 pages, 7 figures. Submitted. Version 2: extended comments on the relation to Lerman's and Whitehouse's work on Menger curvatures

References in corpus (3)

• High-Dimensional Menger-Type Curvatures-Part II: d-Separation and a Menagerie of Curvatures
• A generalization of Reifenberg's theorem in $R^3$
• On the smoothness of Hölder-doubling measures

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• Integral Menger curvature for sets of arbitrary dimension and codimension
• Minimal Hölder regularity implying finiteness of integral Menger curvature
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• Least squares approximations of measures via geometric condition numbers
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• Menger curvatures and $C^{1,\alpha }$ rectifiability of measures
• 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