The Euler-Lagrange equation and heat flow for the Möbius energy

Abstract

We prove the following results: A unique smooth solution exists for a short time for the heat equation associated with the Möbius energy of loops in a euclidean space, starting with any simple smooth loop. A critical loop of the energy is smooth if it has cube‐integrable curvature. Combining this with an earlier result of M. Freedman, Z. Wang, and the author, we show that any local minimizer of the energy must be smooth. Circles are the only two‐dimensional critical loops with cube‐integrable curvature. The technique also applies to a family of other knot energies. Similar problems are open for energies of surfaces or, more generally, for embedded submanifolds in a fixed Riemannian manifold. © 2000 John Wiley & Sons, Inc.

Does Finite Knot Energy Lead to Differentiability?

“…1.1]. One motivation to study such functionals is to find a "nicer", that is, less entangled shape for a given knot in order to determine its knot type, e. g. by following the negative gradient flow of the knot energy up to a local minimum [He00]. By claiming self-repulsion one hopes 1 not to run into the danger of leaving the ambient RWTH Aachen, Institut für Mathematik, Templergraben 55, 52062 Aachen, Germany.…”

How nice are critical knots? Regularity theory for knot energies

“…For the Möbius-energy E 2,1 , we could establish a corresponding result that-using sophisticated methods from harmonic analysis-improves previous results by Freedman, He, Wang [1994] and He [2000]. In light of the technical difficulties arising here we expect these situation for intM (7/3,2) and TP (4,2) to be much more involved.…”

Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α ∈ [2, 3)

“…A couple of years after the joint paper [19], He published his inspiring investigation on the Euler-Lagrange equation and the heat flow associated to the Möbius Energy [30]. As to smoothness of critical points, he treats the special case α = 2 of Theorem 3.23 above.…”

“…As in [30], we have to restrict to test functions in the orthogonal complement ofγ in order to derive a weak Euler-Lagrange equation (3.11) that allows to separate the highest-order terms. The tedious "remainder term" M (α ) which essentially forms a product of derivatives of γ with shifted arguments is thoroughly treated in Lemmata 3.20 and 3.21, filling one of He's major gaps.…”