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

Abstract

We prove the following results: 1 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. 2 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. 3 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.