The Euler-lagrange Equation and Heat Flow for the M Obius Energy

We prove the following results: (1) A unique smooth solution exists for short time for the heat equation associated with the MM obius 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, 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 in general, for embedded submanifolds in a xed Riemannian manifold.