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

We prove the following results: 1. A unique smooth solution exists for a short time for the heat equation a ssociated with the Mobius energy of loops in a euclidean space, starting wi th any simple smooth loop. 2. A critical loop of the energy is smooth if it has cube-integrable curvat ure. 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 prob lems are open for energies of surfaces or, more generally, for embedded sub manifolds in a fixed Riemannian manifold. (C) 2000 John Wiley & Sons, Inc.