Euler-Arnold equations in sub-Riemannian geometry on the Teichmüller space and curve

We consider the group of orientation-preserving diffeomorphisms of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions, which are Ehresmann connections with respect to a projection to the smooth universal Teichmüller space and the universal Teichmüller curve associated to the space of normalized univalent functions. We find equations for the normal sub-Riemannian geodesics with respect to the pullback of the Kählerian metrics, namely, the Velling-Kirillov metric on the class of normalized univalent functions and the Weil-Petersson metric on the universal Teichmüller space. The geodesic equations are sub-Riemannian analogues of the Euler-Arnold equation and they lead to the CLM, KdV and other known non-linear PDEs.