Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth (Q2790734)

After recalling the Coulomb potential of an equidistributed charge on a curve and the Möbius energy (knot energy), the authors prove that every critical curve of this Möbius energy is \(C^\infty\). This is an extension of a result for the minimizers of the Möbius energy due to \textit{M. H. Freedman} et al. [Ann. Math. (2) 139, No. 1, 1--50 (1994; Zbl 0817.57011)]. These authors have shown that there are minimizers for this knot energy within every prime knot class and these are in fact of class \(C^{1,1}\). The regularity argument in the above paper relies on both the Möbius invariance of the energy and the local minimality of the considered curve. In the paper under review, the authors provide a new regularity argument based on the theory of fractional harmonic maps. In fact, they show that even only critical points of the Möbius energy are of class \(C^\infty\) under the condition that the Möbius energy \(E^{(2)}(\gamma)\) of the curve \(\gamma\) is finite. After obtaining the first variation of the Möbius energy, and thus obtaining the corresponding Euler-Lagrange equations, the authors prove their Theorem 1, showing that the stationary points are smooth. Further, in Theorem II, they obtain an explicit expression for the first variation of the energy \(E^{(2)}(\gamma)\). Next, in Theorem III, they obtain the initial regularity and, in Theorem IV, the bootstrapping.