UNIVERSAL TEICHM ULLER SPACE.pdf

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UNIVERSAL TEICHM ULLER SPACE

Annales Academi Scientiarum FennicSeries A. I. MathematicaVolumen 18, 1993, 377{393ON THE TANGENT SPACE TO THEUNIVERSAL TEICHMULLER SPACESubhashis NagThe Institute of Mathematical Sciences, C.I.T. CampusMadras 600 113, India; nag@imsc.ernet.inAbstract. We nd a remarkably simple relationship between the following two models ofthe tangent space to the universal Teichmuller space:(1) The real-analytic model consisting of Zygmund class vector elds on the unit circle;(2) The complex-analytic model comprising 1-parameter families of schlicht functions on theexterior of the unit disc which allow quasiconformal extension.Indeed, the Fourier coecients of the vector eld in (1) turn out to be essentially the sameas (the rst variations of) the corresponding power series coecients in (2).These identities have many applications; in particular, to conformal welding, to the almostcomplex structure of Teichmuller space, to study of the Weil{Petersson metric, to variationalformulas for period matrices, etc. These utilities are explored.1. IntroductionLet  denote the open unit disc, and S1 = @. Two classic models of theuniversal Teichmuller space T (1) = T () are well-known (see [6], [7]):(a) the real-analytic model containing all (Mobius-normalised) quasisymmet-ric homeomorphisms of the unit circle S1 ;(b) the complex-analytic model comprising all (normalised) schlicht functionson the exterior of the disc:? = z 2 bC : jzj 1 = bC ( [ S1)which allow quasiconformal extension to the whole of bC (the Riemann sphere).The connection between them is via the rather mysterious operation called\conformal welding (see [5], and below). Nevertheless, at the in nitesimal level,the above models have an amazingly simple relationship that forms the basis forthis paper. Indeed, the kth Fourier coecient of the vector eld representinga tangent vector in model (a), and the ( rst variation of) the kth power seriescoecient representing the same tangent vector in model (b), turn out

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