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

TL;DR

This paper proves that the energy functional on the Virasoro-Bott group with the $L^2$-metric has no local minima, implying solutions to the KdV-equation do not minimize energy locally.

Contribution

It establishes that the energy functional on the Virasoro-Bott group with the $L^2$-metric has no local minima, linking geometric analysis with the KdV-equation.

Findings

Energy functional has no local minima on the Virasoro-Bott group.

Solutions to KdV do not define locally length-minimizing paths.

The result connects geometric properties with integrable PDEs.

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 don't define locally length-minimizing paths.