The energy functional on the Virasoro–Bott group with the L 2-metric has no local minima

Abstract

The geodesic equation for the right invariant L 2-metric (which is a weak Riemannian metric) on each Virasoro–Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular, solutions of KdV do not define locally length-minimizing paths.