Sub-Riemannian structures corresponding to K\"ahlerian metrics on the universal Teichmueller space and curve

Abstract

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 Teichm\"uller space and the universal Teichm\"uller curve associated to the space of normalized univalent functions. We find formulas for the normal geodesics with respect to the pullback of the invariant K\"ahlerian metrics, namely, the Velling-Kirillov metric on the class of normalized univalent functions and the Weil-Petersson metric on the universal Teichm\"uller 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.