Towards a regularity theory for integral Menger curvature. (English) Zbl 1362.53007
Summary: We generalize the notion of integral Menger curvature introduced by O. Gonzalez and J. H. Maddocks [Proc. Natl. Acad. Sci. USA 96, No. 9, 4769–4773 (1999; Zbl 1057.57500)] by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies \(\text{intM}^{(p,q)}\). We classify finite-energy curves in terms of Sobolev-Slobodeckii 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, \(\text{intM}^{(p,q)}\) 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 \(C^{\infty}\)-smoothness of critical points.
MSC:
| 53A04 | Curves in Euclidean and related spaces |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
Keywords:
knot energy; Menger curvature; integral Menger curvature; regularity; fractional Sobolev spaces; fractional seminormsCitations:
Zbl 1057.57500References:
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