Kähler geometry of the universal Teichmüller space and coadjoint orbits of the Virasoro group. (English. Russian original) Zbl 1351.32023

Sergeev, A. G.

Proc. Steklov Inst. Math. 253, 160-185 (2006); translation from Tr. Mat. Inst. Steklova 253, 175-203 (2006).

Summary: 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 \(\mathrm{Diff}_{+}(S^{1})/\mathrm{M\"ob}(S^{1})\) and \(\mathrm{Diff}_{+}(S^{1})/S^{1}\) of the diffeomorphism group \(\mathrm{Diff}_{+}(S^{1})\) 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).

MSC:

32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
58B20	Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
17B68	Virasoro and related algebras

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