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 on each Virasoro-Bott group is equivalent to the Korteweg-de Vries equation. The main purpose of this paper is to prove that the energy functionals (when restricted to paths with fixed endpoints) for the Virasoro-Bott group and for the diffeomorphism group of \(\mathbb{R}\) have no local minima. In particular, the author gets the same result for the length functional. Given any curve there is a close by curve with the same endpoints that uses less energy. Hence no geodesic is locally energy or length minimizing.
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