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

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 don’t define locally length-minimizing paths.