Towards a regularity theory for integral Menger curvature

2013/08/12 by Simon Blatt, Blatt, Simon, Philipp Reiter +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Topology (math.GT) #math.AP #math.GT

paper · pdf · doi:10.48550/arxiv.1308.2499

32 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1208.3605

arxiv created 2013/08/12 · arxiv updated 2013/08/13

Abstract

We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks 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-Slobodeckij 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. Consequently, 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.

Citations

• Regularity theory for tangent-point energies: The non-degenerate sub-critical case

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