Integral Menger Curvature Research Articles

On the analyticity of critical points of the generalized integral Menger curvature in the Hilbert case

We prove the analyticity of smooth critical points for generalized integral Menger curvature energies intM(p,2), with p∈(73,83), subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical C1-curves γ:R/Z→Rn of generalized integral Menger curvature intM(p,2) subject to a fixed length constraint are not only C∞ but also analytic. Our approach is inspired by analyticity results on critical points for O’Hara’s knot energies based on Cauchy’s method of majorants and a decomposition of the first variation. The main new idea is an additional iteration in the recursive estimate of the derivatives to obtain a sufficient difference in the order of regularity.

A speed preserving Hilbert gradient flow for generalized integral Menger curvature

Abstract We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Characterizations of countably $n$-rectifiable radon measures by higher-dimensional Menger curvatures

We provide a characterization of countably n-rectifiable measures in terms of $\sigma$-finiteness of the integral Menger curvature. We also prove that a finiteness condition on pointwise Menger curvature can characterize rectifiability of Radon measures. Motivated by the partial converse of Meurer's work by Kolasiński we prove that under suitable density assumptions there is a comparability between pointwise-Menger curvature and the sum over scales of the centered $\beta$-numbers at a point.

Characterization of Sobolev-Slobodeckij spaces using geometric curvature energies

We give a new characterization of Sobolev-Slobodeckij spaces W^{1+s,p} for n/p<1+s, where n is the dimension of the domain. To achieve this we introduce a family of curvature energies inspired by the classical concept of integral Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij space if and only if it is in L^p and the appropriate energy is finite.

Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

In this work we show that an $n$-dimensional Borel set in Euclidean $N$-space with finite integral Menger curvature is $n$-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null set in the sense of Hausdorff measure. This generalises Leger's rectifiability result for one-dimensional sets to arbitrary dimension and co-dimension. In addition, we characterise possible integrands and discuss examples known from the literature. Intermediate results of independent interest include upper bounds of different versions of P. Jones's $\beta$-numbers in terms of integral Menger curvature without assuming lower Ahlfors regularity, in contrast to the results of Lerman and Whitehouse.

Towards a regularity theory for integral Menger curvature

We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks (14) by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies intM (p;q) . We classify finite-energy curves in terms of Sobolev-Slobodecki˘i spaces. Moreover, restricting to the range of parameters leading to a sub-critical Euler-Lagrange equation, we prove existence of minimizers within any knot class via a uniform bi-Lipschitz bound. Consequemtly, intM (p;q) is a knot energy in the sense of O'Hara. Restricting to the non-degenerate sub-critical case, a suitable decomposition of the first variation allows to establish a bootstrapping argument that leads to C 1 -smoothness of critical points.

How nice are critical knots? Regularity theory for knot energies

In this note we report on some recent developments in geometric knot theory which aims at finding links between geometric properties of a given knotted curve and its knot type. The central object of this field are so-called knot energies which are defined on closed embedded curves.First we present three important examples of two-parameter knot energy families, namely O'Hara's energies, the (generalized) integral Menger curvature, and the (generalized) tangent- point energies.Subsequently we outline the main steps that lead to C inf -regularity of stationary points- especially minimizers-in the non-degenerate sub-critical range of parameters.Particular attention is devoted to the appearing parallels between these energies which, surprisingly at first glance, are quite similar from an analyst's perspective.

On some knot energies involving Menger curvature

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.

A note on integral Menger curvature for curves

AbstractFor continuously differentiable embedded curves γ, we will give a necessary and sufficient condition for the boundedness of integral versions of the Menger curvature. We will show that for p &gt; 3 the integral Menger curvature \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {M}_p(\gamma )$\end{document} is finite if and only if γ belongs to the Sobolev Slobodeckij space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^{2-\frac{2}{p},p}(\mathbb {R} / \mathbb {Z}, \mathbb {R}^n)$\end{document}. The quantity \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {J}_p (\gamma )$\end{document} – defined by taking the supremum of the p‐th power of Menger curvature with respect to one variable and then integrating over the remaining two – is finite for p &gt; 2, if and only if γ belongs to the space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^{2-\frac{1}{p},p}$\end{document}.

Minimal Hölder regularity implying finiteness of integral Menger curvature

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 $${C^{1,\alpha_0}}$$ injective curve and higher dimensional embedded manifolds with unbounded curvature.

Sharp boundedness and regularizing effects of the integral Menger curvature for submanifolds

We show that embedded and compact C1 manifolds have finite integral Menger curvature if and only if they are locally graphs of functions belonging to 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.

Tangency properties of sets with finite geometric curvature energies

We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature \begin{equation*} \newcommand{

Integral Menger curvature for surfaces

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.

Regularizing and self-avoidance effects of integral Menger curvature

We investigate geometric curvature energies on closed curves involving integral versions of Menger curvature. In particular, we prove geometric variants of Morrey-Sobolev and Morrey-space imbedding theorems, which may be viewed as counterparts to respective results on one-dimensional sets in the context of harmonic analysis.

Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

We prove isotopy finiteness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined on the class of compact, embedded m-dimensional Lipschitz submanifolds in R n . That is, there are only finitely many isotopy types of such submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. Moreover, we establish C 1 -compactness: any sequence of submanifolds with uniformly bounded en- ergy contains a subsequence converging inC 1 to a limit submanifold with the same energy bound. In addition, we show that all geometric curvature energies under consideration are lower semicontinuous with respect to Hausdor-converge nce, which can be used to min- imise each of these energies within prescribed isotopy classes.