Integral Menger curvature for surfaces

TL;DR

This paper introduces integral Menger curvature for surfaces, establishing regularity, scale-invariance, and existence results, and providing a new geometric Morrey-Sobolev embedding theorem for nonsmooth surfaces.

Contribution

It extends the concept of integral Menger curvature to nonsmooth surfaces, proving regularity, scale invariance, and existence of minimizers, with a new optimal Hölder exponent.

Findings

Surfaces with finite integral Menger curvature are uniformly regular and nearly flat at small scales.

The paper establishes a new Morrey-Sobolev embedding with optimal Hölder exponent.

Existence of curvature-minimizing and area-minimizing surfaces under energy bounds is proven.

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 H\"{o}lder 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.