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

Abstract

Motivated by the Coulomb potential of an equidistributed charge on a curve, Jun O’Hara introduced and investigated the first geometric knot energy, the Möbius energy. We prove that every critical curve of this Möbius energy is of class $C^\infty$ and thus extend the corresponding result due to Freedman, He, and Wang for minimizers of the Möbius energy. In contrast to the techniques used by Freedman, He, and Wang, our methods do to not use the Möbius invariance of the energy, but rely on purely analytic methods motivated from a formal similarity of the Euler-Langrange equation to the half harmonic map equation for the unit tangent.