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THE UNIVERSAL PROPERTIES OF TEICHMULLERÄ SPACES VLADIMIR MARKOVIC AND DRAGOMIR SARI· C¶ Abstract. We discuss universal properties of general TeichmÄullerspaces. Our topics include the TeichmÄullermetric and the Kobayashi metric, extremality and unique extremality of quasiconformal mappings, biholomorphic maps be- tween TeichmÄullerspace, earthquakes and Thurston boundary. 1. Introduction Today, TeichmÄullertheory is a substantial area of mathematics that has inter- actions with many other subjects. The bulk of this theory is focused on studying TeichmÄullerspaces of ¯nite type Riemann surfaces. In this article we survey the theory that investigates all TeichmÄullerspaces regardless of their dimension. We aim to present theorems (old and recent) that illustrate universal properties of TeichmÄullerspaces. TeichmÄullerspaces of ¯nite type Riemann surfaces (or just ¯nite Riemann sur- faces) are ¯nite-dimensional complex manifolds with rich geometric structures. Te- ichmÄullerspaces of in¯nite type Riemann surfaces are in¯nite-dimensional Banach manifolds whose geometry di®ers signi¯cantly from the ¯nite case. However, some statements hold for both ¯nite and in¯nite cases. The intent is to describe these universal properties of all TeichmÄullerspaces and to point out to di®erences between ¯nite and in¯nite cases when these are well understood. The following is the list of topics covered. In the second section we briefly in- troduce quasiconformal maps and mention their basic properties. Then we proceed to give the analytic de¯nition of TeichmÄullerspaces, regardless whether the un- derlying Riemann surface is of ¯nite or in¯nite type. We de¯ne the TeichmÄuller metric and introduce the complex structure on TeichmÄullerspaces. Next we dis- cuss the Kobayashi metric, the tangent space and the barycentric extensions. In the third section we consider the geometry of TeichmÄullerspaces. We cover the Reich-Strebel inequality and the TeichmÄullertheorem, the Finsler structure, the universal TeichmÄullerspace, extremal and uniquely extremal quasiconformal maps. In section four we consider biholomorphic maps between TeichmÄullerspaces and give a short proof that the modular group is the full group of biholomorphic maps of the TeichmÄullerspace of a ¯nite surface following [20]. In section ¯ve we consider earthquakes and bending on in¯nite surfaces, and we introduce Thurston boundary for general TeichmÄullerspaces. Date: January 9, 2009. 1991 Mathematics Subject Classi¯cation. 30F60. 1 2 VLADIMIR MARKOVIC AND DRAGOMIR SARI· C¶ 2. The TeichmullerÄ space: definition, the TeichmullerÄ and Kobayashi metric, the complex structure and barycentric extension We start with basic de¯nitions. Let M be a Riemann surface. The Uniformisa- tion theorem states that the universal covering of M is either the complex plane C, the Riemann sphere Cb = C [ f1g or the upper half-plane H. The complex plane C, the once punctured complex plane C n f0g and the torus T is the short list of Riemann surfaces covered by C. The Riemann sphere Cb is the only Riemann surface whose universal covering is Cb . The TeichmÄullerspaces of Cb , C and C n f0g consist of a single point, while the TeichmÄullerspace of the torus T is biholomor- phic to the upper half-plane H and isometric to the hyperbolic plane (for example, see [41]). Thus, the TeichmÄullerspace of a Riemann surface M whose universal covering is either Cb or C is well understood. We focus on the case when H is the universal covering. A quasiconformal map f : H ! H is an orientation preserving homeomorphism ¹ which is absolutely continuous on lines and which satis¯es k@f=@fk1 < 1. The Beltrami coe±cient ¹ = @f=@f¹ of a quasiconformal map f is de¯ned almost every- where and it satis¯es k¹k1 < 1. The quasiconformal constant K(f) of f is given 1+k¹k1 by K(f) = . Note that k¹k1 < 1 if and only if K(f) < 1. 1¡k¹k1 Given a measurable function ¹ on H such that k¹k1 < 1, then there exists a quasiconformal map f : H ! H such that ¹ = @f=@f¹ . The quasiconformal map f is unique up to post-composition by a MÄobiusmap preserving H (see [4]). 2.1. De¯nition of the TeichmÄullerspace. From now on we assume that the universal covering of a Riemann surface M is the upper half-plane H. We identify the hyperbolic plane with the upper half-plane H equipped with the metric ½(z) = jdzj 2y , where z = x + iy 2 H. The universal covering map ¼ : H ! M induces a hyperbolic metric on M. The Riemann surface M is said to be hyperbolic. Let G be a Fuchsian group acting on the upper half-plane H such that M is conformal and isometric to H=G. (The group G is unique up to conjugation by a MÄobiusmap ¯xing H.) Let P SL2(R) denote the subgroup of the MÄobiusgroup which ¯xes the upper half-plane H. De¯nition 2.1. Let M be a hyperbolic surface. Let G be a Fuchsian group such that M is isomorphic to H=G. The TeichmÄuller space T (M) of M consists of equivalence classes of quasiconformal maps f : H ! H which satisfy the following condition ¡1 (1) f ± γ ± f 2 P SL2(R); for all γ 2 G. Two such quasiconformal maps f1; f2 : H ! H are equivalent if their extensions to the extended real line Rb = R [ f1g agree up to a post-composition by a MÄobiusmap, i.e. f1 is equivalent to f2 if f1jRb = ¯ ±f2jRb for some ¯ 2 P SL2(R). Remark. The de¯nition of T (M) depends on the choice of the Fuchsian group G. Since G is unique up to conjugation by an element of P SL2(R) it is easy to check that all subsequent de¯nitions are independent of this choice. We denote by [f] the equivalence class of the quasiconformal map f : H ! H satisfying the invariance property (1). Then [f] 2 T (M). THE UNIVERSAL PROPERTIES OF TEICHMULLERÄ SPACES 3 Remark. In the above de¯nition, we could replace quasiconformal maps of H with quasisymmetric maps of Rb which satisfy the invariance property (1) on Rb . This follows from the Douady-Earle barycentric extension [12]. Remark. The map f : H ! H which satis¯es the invariance property (1) projects to ^ a quasiconformal map f : M ! M1, where M = H=G and M1 is a Riemann surface ¡1 whose covering Fuchsian group is fGf . The condition that f1jRb = ¯ ± f2jRb is ^ ^ equivalent to the property that the projections f1 and f2 map M onto the same surface and that they are isotopic through a bounded quasiconformal isotopy. The last statement was proved by Earle and McMullen [22] using the Douady-Earle extension. 2.2. The TeichmÄullermetric. The TeichmÄullerspace T (M) carries a natural metric de¯ned as follows. De¯nition 2.2. Let [f]; [g] 2 T (M). The TeichmÄuller distance between [f] and [g] is given by 1 ¡1 d([f]; [g]) = inf log K(g1 ± f1 ): g12[g];f12[f] 2 It is easy to check that this distance is in fact a metric. The space (T (M); d) is a complete and non-compact metric space. 2.3. The complex structure on the TeichmÄullerspace. The TeichmÄuller space is equipped with a natural complex structure as follows. Let [f] 2 T (M) ¹ and let ¹ = @f=@f be the Beltrami coe±cient of f. Then k¹k1 < 1 and γ0(z) (2) ¹(z) = ¹(γ(z)) γ0(z) for z 2 H and γ 2 G. Let L = C n H be the lower half-plane. Given a Beltrami coe±cient ¹ on H which satis¯es (2), we de¯ne the Beltrami coe±cient ¹b on Cb by ¹b(z) = ¹(z) for z 2 H, and ¹b(z) = 0 for z 2 L. Then ¹b satis¯es (2) for all z 2 Cb . There exists a quasiconformal map f : Cb ! Cb whose Beltrami coe±cient is ¹b (see [4] for the solution of the Beltrami equation @f¹ = ¹b ¢ @f). Moreover, f is unique up to a post-composition by a MÄobiusmap of the Riemann sphere Cb , it is conformal in L and it satis¯es the invariance relation (1) in Cb . Denote this map by f ¹b. Let g be a locally injective holomorphic map de¯ned in a domain on Cb . Then the Schwarzian derivative S(g) of g is given by g000 3³g00 ´2 S(g) = ¡ : g0 2 g0 We recall that the Schwarzian derivative measures by how much a holomorphic map distorts cross-ratios of four points (see [34, Section 6.1]). In particular, the Schwarzian derivative of a MÄobiusmap is zero. If we apply the Schwarzian derivative to f ¹b in L, then we obtain a holomorphic map S(f ¹b) on L which satis¯es (3) (S(f ¹b) ± γ)(z)(γ0(z))2 = S(f ¹b)(z) and ¹b ¡2 (4) sup jS(f )(z)½L (z)j < 1 z2L 4 VLADIMIR MARKOVIC AND DRAGOMIR SARI· C¶ jdzj for z 2 L and γ 2 G, where ½L(z) = 2jyj is the Poincar¶emetric on L (see [6], [29] or [34]). Let BL(G) be the Banach space of all holomorphic maps à : L ! C which satisfy 0 2 ¡2 ¡2 Ã(γ(z))γ (z) = Ã(z), for z 2 L and γ 2 G, and kýL k1 = supz2L jÃ(z)½L (z)j < ¡2 ¹^ 1, where kýL k1 is the norm on BL(G). Note that S(f ) 2 BL(G). If a quasiconformal map f : H ! H satis¯es the invariance property (1), then the Beltrami coe±cient ¹ = @f=@f of f satis¯es the invariance property (2) (and k¹k1 < 1). Conversely, given measurable ¹ : H ! C such that k¹k1 < 1 and (2) holds for ¹, then there exists a quasiconformal map f : H ! H whose Beltrami coe±cient is ¹ and which satis¯es (1) (see [4]).
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