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

TL;DR

This paper explores the sub-Riemannian geometry of the universal Teichmüller space and related structures, deriving geodesic equations that connect to well-known nonlinear PDEs like KdV.

Contribution

It derives explicit formulas for normal geodesics in the context of Kählerian metrics on the universal Teichmüller space, linking geometric structures to integrable PDEs.

Findings

Formulas for normal geodesics under Kählerian metrics

Connections between sub-Riemannian geodesics and nonlinear PDEs

Extension of geometric structures to the Virasoro-Bott group

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.