K\"ahler Geometry of the Universal Teichm\"uller Space and Coadjoint Orbits of the Virasoro Group

A. G. Sergeev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 175-203

Résumé

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).