Towards a regularity theory for integral Menger curvature

Abstract

Abstract. 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-Slobodeckiȋ 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 ∞ -smoothness of critical points.

Integro-Differential Harmonic Maps into Spheres

“…For several of these curvature energies, regularity even of minimizers is not understood, and arguments as in [19] seem not to work, since also the invariance class is not known. See [7,9,31,32]. Thus, the present work is also intended to deliver a framework which hopefully will lead to substantial progress in this area.…”

Dynamics of embedded curves by doubly-nonlocal reaction–diffusion systems

“…This insight allows us to establish complexity bounds for a rather broad class of kernels K(u, v), and it also provides the impetus to introduce generalizations of such complexity measures. In this sense, our work complements that strand of geometric knot theory that emphasizes the importance of taking a more analytic approach to (1) and its variants [14,15,16]. For instance, the relative importance of harmonic analysis vis-à-vis Möbius invariance has been observed when studying regularity of extremal embeddings [17].…”

How nice are critical knots? Regularity theory for knot energies

“…gives rise to a full two-parameter family of generalized integral Menger curvature functionals [Blatt & Reiter, 2013b] intM…”

“…Theorem (Stationary points are smooth [Blatt & Reiter, 2013a, 2013b). Any stationary point of E α,1 , intM (p,2) , and TP (p,2) in the non-degenerate sub-critical case with respect to fixed length, being injective and parametrized by arc-length, is C ∞ -smooth.…”