Alexander Vasil'ev: Sub-Riemannian structures corresponding to Kaehlerian metrics on the universal Teichmueller space and curve

We consider the group of sense-preserving diffeomorphisms Diff S^1 of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions chosen to be Ehresmann connections with respect to a projection to the smooth universal Teichmueller space and the universal Teichmueller curve associated to the space of normalized univalent functions. We find formulas for the normal geodesics with respect to the pullback of the invariant Kaehlerian metrics, namely, the Velling-Kirillov metric on the class of normalized univalent functions and the Weil-Petersson metric on the universal Teichmueller space. The geodesic equations are sub-Riemannian analogues of the Euler-Arnold equation and lead to the CLM, KdV, and other known non-linear PDE.