Kähler Geometry of the Universal Teichmüller Space and Coadjoint Orbits of the Virasoro Group

The Kähler geometry of the universal Teichmüller space and related infinite-dimensional Kähler manifolds is studied. The universal Teichmüller space $\mathcal T$ may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The classical Teichmüller spaces $T(G)$, where $G$ is a Fuchsian group, are contained in $\mathcal T$ as complex Kähler submanifolds. The homogeneous spaces $\text {Diff}_+(S^1)/\text {M\"ob}(S^1)$ and $\text {Diff}_+(S^1)/S^1$ of the diffeomorphism group $\text {Diff}_+(S^1)$ of the unit circle are closely related to $\mathcal T$. They are Kähler Frechet manifolds that can be realized as coadjoint orbits of the Virasoro group (and exhaust all coadjoint orbits of this group that have the Kähler structure).