Lectures on Universal Teichmuller Space

EMS Series of Lectures in Mathematics Edited by Andrew Ranicki (University of Edinburgh, U.K.) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim, Uhlenbeck Compactness Torsten Ekedahl, One Semester of Elliptic Curves Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration Koen Thas, A Course on Elation Quadrangles Benoît Grébert and Thomas Kappeler, The Defocusing NLS Equation and Its Normal Form Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink and Jürgen Sander, Four Faces of Number Theory

Armen Sergeev

Lectures on Universal Teichmüller Space

Author: Armen Sergeev Steklov Mathematical Institute Gubkina 8 119991 Moscow Russia E-mail: [email protected]

2010 Mathematics Subject Classification: Primary: 58B20, 58B25, 58B34; Secondary: 53C55, 53D50 Key words: Teichmüller spaces, conformal maps, quasisymmetric homeomorphisms, Kähler manifolds, geometric quantization, noncommutative geometry

ISBN 978-3-03719-141-5 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2014

Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland



Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org

Typeset using the author’s TEX files: I. Zimmermann, Freiburg, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface

Generalizing the theory of classical Teichmüller spaces of compact Riemann surfaces, Lipman Bers introduced the notion of universal Teichmüller space (cf. [4], [5], [6]). It is an infinite-dimensional complex manifold containing all classical Teichmüller spaces as complex submanifolds which explains the name “universal Teichmüller space”. This space, denoted by T , is defined as the quotient of the space of quasisymmetric homeomorphisms of the unit circle modulo Möbius group of fractional-linear automorphisms of the unit disk. Recall that a homeomorphism of the circle is called quasisymmetric if it can be extended into the disk as a quasiconformal map. Apart from the classical Teichmüller spaces, T contains the space  of diffeomorphisms of the circle, considered up to Möbius transformations. Both spaces T and  are the important examples of infinite-dimensional complex manifolds, having a rich geometric structure. Both of them can be embedded into the “lower hemisphere” of an infinite-dimensional Grassmann manifold and so may be considered as infinitedimensional analogues of the unit disk. In particular, the space  can be provided with a Kähler metric, invariant under the action of diffeomorphisms of the circle. This metric has a correctly defined Ricci curvature which is given in a suitable basis by an infinite diagonal matrix. Similar to the unit disk, this curvature is negative meaning that the coefficient of the leading term is equal to  26 . Precisely, this number 26 arises 12 in the bosonic string theory as the “critical dimension” of the Minkowski space where the strings “live”. In fact, relations with string theory were our main motivation to study the spaces  and T . Recall that  is the quotient of the diffeomorphism group of the circle modulo Möbius group and the group of diffeomorphisms acts on the strings by the reparameterization. This explains a relation of  with the theory of smooth strings. The same role is played by the space T for the theory of non-smooth strings. Again T is the quotient of the group of quasisymmetric homeomorphisms of the circle modulo Möbius group, acting on half-differentiable strings by the reparameterization. We note that there are no physical reasons to restrict ourselves to smooth strings, rather such a restriction is explained by the mathematical convenience of working with smooth objects. From our point of view it is much more natural to choose the Sobolev space of half-differentiable vector-functions for the phase manifold of the string theory. Such a choice is motivated by the fact that this space is the largest among all Sobolev spaces on which the symplectic form of string theory is still correctly defined. (We discuss a relation between the universal Teichmüller space and string theory in the afterword to this book.) To quantize the string theory with the phase manifold given by the space of halfdifferentiable vector-functions, we have to quantize the space T . Its quantization is considered in the second part of the book. In contrast with the case of the space , which

vi

Preface

can be quantized in frames of the conventional Dirac scheme (cf. [27]), the quantization of T requires dealing with “non-smooth” objects to which the Dirac scheme does not apply. In order to quantize the universal Teichmüller space, we have to use a completely different approach motivated by considerations from the non-commutative geometry. This book is based on the lecture notes of the course delivered by the author to the students of the Scientific Educational Center of Steklov Mathematical Institute during the spring semester of the year 2011 (the Russian edition of the lecture notes was published by the Steklov Institute in 2013 [26]). Despite the fact that this book is a lecture course, divided into separate lectures, we think of these lectures more like of separate topics rather than “real” lectures (this explains the difference in their volumes). We have included in this book many problems which have been proposed to the listeners of the course. They are not divided into exercises and “difficult” problems, we hope that a reader will understand the difference by himself. We only guarantee that among them there are no unsolved problems. At the end I want to express my gratitude to Alastair Fletcher and Manfred Karbe who contributed a lot to the publication of this book in the European Lecture Notes Series. I am also grateful to Irene Zimmermann for the excellent typesetting of the text. Last but not least I want to thank all listeners of the course for their questions and remarks Moscow, July 2014

Armen Sergeev

Contents

Preface

v

1

Quasiconformal maps 1.1 Lecture I. Definition of quasiconformality . . . . . . . . . . . . . . . 1.2 Lecture II. Uniqueness and existence theorems . . . . . . . . . . . . 1.3 Lecture III. Quasisymmetric homeomorphisms . . . . . . . . . . . .

1 1 5 12

2

Universal Teichmüller space 2.1 Lecture IV. Definition of the universal Teichmüller space . . . . . . . 2.2 Lecture V. Properties of the universal Teichmüller space . . . . . . . .

19 19 24

3

Subspaces of universal Teichmüller space 3.1 Lecture VI. Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . 3.2 Lecture VII. Classical Teichmüller spaces . . . . . . . . . . . . . . . 3.3 Lecture VIII. The space of diffeomorphisms . . . . . . . . . . . . . .

37 37 44 48

4

Grassmann realization of the universal Teichmüller space 4.1 Lecture IX. The action of quasisymmetric homeomorphisms on the Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Lecture X. Grassmann realization of the space T . . . . . . . . . . .

55

5

Quantization of space of diffeomorphisms 5.1 Lecture XI. Quantization of classical systems by Dirac . . . . . . . . 5.2 Lecture XII. Quantization of the extended system . . . . . . . . . . .

69 69 72

6

Quantization of Teichmüller space 6.1 Lecture XIII. Quantization by Connes . . . . . . . . . . . . . . . . . 6.2 Lecture XIV. Quantization of the universal Teichmüller space . . . . .

85 85 90

7

Instead of an afterword. Universal Teichmüller space and string theory

93

8

Problems

95

9

Bibliographical comments

99

55 64

Bibliography

101

Index

103

1 Quasiconformal maps

The theory of universal Teichmüller space is based on general properties of quasiconformal maps, presented in this chapter. We give three different definitions of quasiconformal maps of the complex plane – as maps with bounded dilatation (Section 1.1.2), as solutions of Beltrami equation (Section 1.1.3), and in terms of conformal moduli (Section 1.3.2). We prove two main theorems on the existence (Section 1.2.2) and uniqueness (Section 1.2.1) of quasiconformal maps. Quasiconformal homeomorphisms of the disk onto itself form a group. All maps from this group are continuous up to the boundary and their boundary values are called quasisymmetric homeomorphisms. An intrinsic description of quasisymmetric homeomorphisms is given by the Beurling–Ahlfors theorem (Section 1.3.3).

1.1 Lecture I. Definition of quasiconformality 1.1.1 Quasiconformal diffeomorphisms. The notion of conformality originates in optics where it means that the section of a round light beam, orthogonal to the axis of the beam, remains round while moving along this axis. The mathematical interpretation of this notion is formulated in the following way. A smooth orientation-preserving map w of the complex plane is conformal at a point z if the map w .z/ W C ! C, tangent to w and given by the formula w .z/./ D w.z/ C

@w @w .z/.  z/ C .z/.N  z/; N @z @zN

sends circles with centers at the point z to circles centered at the point w.z/. In other words, the differential dw.z/ is a non-degenerate real-linear map, represented as the composition of a rotation and a dilatation. By analogy with this definition, we can give the following geometric definition of smooth quasiconformal maps. A diffeomorphism w of the complex plane, preserving the orientation, is called quasiconformal if the tangent map w .z/, sending circles with centers at a point z to ellipses centered at the point w.z/, satisfies the following condition: the eccentricities of these ellipses 1 are bounded by a constant which does not depend upon the point z. 1 The eccentricity of an ellipse is the ratio of the length of its large half-axis to the length of its small half-axis.

2

1 Quasiconformal maps

Let us try to convert this geometric definition into an analytic one. Suppose that w is an orientation-preserving diffeomorphism of the complex plane. Denote by @ w.z/ D lim

!0

w.z C e i /  w.z/ e i

its derivative in the direction  at a point z. A simple calculation shows that this derivative is equal to N  e 2i @ w.z/ D @w C @w where @w WD

@w .z/; @z

N WD @w .z/: @w @zN

It follows that N max j@ w.z/j D j@wj C j@wj; 

N > 0: min j@ w.z/j D j@wj  j@wj 

N is positive everywhere because of the positivity of the Jacobian The quantity j@wjj@wj 2 2 N Jw D j@wj  j@wj of the orientation-preserving diffeomorphism w. The eccentricity of the ellipse with center at w.z/ is equal to Dw .z/ D

N max j@ w.z/j j@wj C j@wj : D N min j@ w.z/j j@wj  j@wj

N  0. In particular, the map w is everywhere conformal () Dw  1 () @w Problem 1. Prove that the eccentricity Dw is a conformal invariant of the map w, i.e. Dw .z/ D DhBwBg 1 .g.z// for conformal maps h and g. Definition 1. An orientation-preserving diffeomorphism w W D ! D 0 of a domain D  C in the complex plane onto a domain D 0  C is called K-quasiconformal in the domain D if Dw .z/  K for all z 2 D. In other words, max j@ w.z/j  K min j@ w.z/j 

for all z 2 D.



(1.1)

1.1 Lecture I. Definition of quasiconformality

3

1.1.2 Definition of quasiconformal maps. Let us try to extend the Definition 1, given in the last section, to non-smooth homeomorphisms. The first idea is to define quasiconformal maps as orientation-preserving homeomorphisms w W D ! D 0 which are differentiable almost everywhere in D and satisfy the condition (1.1) for almost all points z 2 D. However, such a definition will deviate from the geometric definition of quasiconformal maps in terms of conformal rectangles given below in Section 1.3.2 (cf. [17], p. 167). We obtain the right definition of quasiconformal maps if we impose instead of the condition of differentiability almost everywhere the following condition of absolute continuity on rectangles. Definition 2. A continuous real-valued function u, defined in a domain D  C, is called absolutely continuous on rectangles (RAC) if for any rectangle … D fx C iy W a  x  b; c  y  d g  D the function x 7! u.x C iy/ is absolutely continuous in x 2 Œa; b for almost all y 2 Œc; d  and the function y 7! u.x C iy/ is absolutely continuous in y 2 Œc; d  for almost all x 2 Œa; b. A complex-valued function w D u C iv, defined in a domain D, has the property (RAC) if both its real part u and imaginary part v have this property. Remark 1. Using standard theorems from the real function theory, it can be proved that the function w with the property (RAC) in a domain D has finite partial derivatives in x and y almost everywhere in D. In particular, its angular derivative @ w exists for every  for almost all w. Definition 3. A homeomorphism w, defined in a domain D and preserving the orientation, is called K-quasiconformal if 1) it has the property (RAC) in D; 2) max j@ w.z/j  K min j@ w.z/j for almost all z 2 D. A homeomorphism w in a domain D is quasiconformal it is K-quasiconformal for some K. 1.1.3 Beltrami equation. Let w W D ! D 0 be a K-quasiconformal map which is differentiable at a point z 2 D. Then the inequality from Definition 3, N j@wj C j@wj max j@ w.z/j  K; D N min j@ w.z/j j@wj  j@wj may be rewritten in the form N [email protected]/j 

K 1 [email protected]/j: K C1

4

1 Quasiconformal maps

Since the Jacobian Jw .z/ is positive at z we have @w.z/ ¤ 0 and we can introduce the function N @w.z/ : .z/ WD @w.z/ This function, defined by construction almost everywhere in D, is called the complex dilatation of the map w. It is measurable (since the function w is continuous) and satisfies the estimate K 1 ess supz2D j.z/j  < 1; K C1 in other words,  belongs to the unit ball in the Banach space L1 .D; C/. Geometrically, .z/ is measuring the eccentricity of the ellipses obtained from the circles with centers at z under the action of the tangent map w .z/. Concretely, this dilatation is equal to 1 C j.z/j : 1  j.z/j If .z/ D 0 then the map w is conformal at z. On the other hand, if .z/ ¤ 0 then the angle 12 arg .z/ determines the direction of maximal stretching of w at z in the sense that for  D 12 arg .z/ the modulus j@ w.z/j of the directional derivative attains its maximum. Definition 4. The equation

N D  @w @w

(1.2)

on a function w in a domain D, where  is a given measurable bounded function in D with kk1 < 1, is called the Beltrami equation. In the case when   0 this equation reduces to the Cauchy–Riemann equation. Remark 2. The function  in the equation (1.2) is called the Beltrami differential since it behaves under conformal changes of variable, according to equation (1.2), as a differential. More precisely, if f is a conformal map defined in a domain D then the function  under the action of f transforms as .f .z// D .z/

f 0 .z/ f 0 .z/

:

Recall that a function ', defined in a domain D, is called the differential of type .m; n/, where m; n 2 Z, if the form ' dz m d zN n remains invariant under conformal changes of variable z. From this point of view the Beltrami differential is a differential of type .1; 1/. We say that a function w is an Lp -solution of the Beltrami equation (1.2) if the derivatives of w of the first order belong to Lp .D/ and the equation (1.2) is satisfied almost everywhere in D. The Beltrami equation can be used to define the quasiconformal maps since we have the following theorem proved in [16], I.4.2.

1.2 Lecture II. Uniqueness and existence theorems

5

Theorem 1. A homeomorphism w, defined in a domain D, is K-quasiconformal if and only if it is an L2 -solution of Beltrami equation (1.2) with Beltrami differential  2 L1 .D; C/, satisfying kk1 < 1. Brief content of Lecture I We have given two definitions of quasiconformal maps of domains in the complex plane C. According to the first definition, an orientation-preserving homeomorphism w defined in a domain D  C is called K-quasiconformal if it is absolutely continuous on rectangles and its directional derivative has the property max j@ w.z/j  K min j@ w.z/j 



for almost all z 2 D. According to the second definition in terms of Beltrami differentials, a homeomorphism w given in a domain D and preserving the orientation, is quasiconformal if and only if it is an L2 -solution of the Beltrami equation N D  @w @w where the function , which is equal almost everywhere to .z/ WD

N @w.z/ ; @w.z/

is called the complex dilatation or Beltrami differential. It belongs to the unit ball in the space L1 .D; C/.

1.2 Lecture II. Uniqueness and existence theorems 1.2.1 The composition of quasiconformal maps. The Uniqueness Theorem. Suppose that we have two quasiconformal maps, defined in a domain D: a map f with complex dilatation f and g with complex dilatation g . Then their composition f B g 1 is again a quasiconformal map as it follows from the next theorem. Theorem 2 (Composition Theorem). The inverse map of a K-quasiconformal map is again K-quasiconformal. The composition of a K1 -quasiconformal map f1 W D ! D 0 and K2 -quasiconformal map f2 W D 0 ! D 00 is a K1 K2 -quasiconformal map. The proof of this assertion is left to the reader as an exercise (to prove it one can use the geometric definition of quasiconformal maps given below in Section 1.3.2).

6

1 Quasiconformal maps

Problem 2. Prove the following composition formula for Beltrami differentials: f .z/  g .z/



@g.z/ f Bg 1 .g.z// D 1  f .z/g .z/ [email protected]/j

2 ;

satisfied for almost all z 2 D. Theorem 3 (Uniqueness Theorem). Let f and g be two quasiconformal maps, defined in a domain D, such that their complex dilatations coincide almost everywhere in D: f D g a.e. Then the maps f Bg 1 and g Bf 1 are conformal. The composition hBf of the map f with an arbitrary conformal map h, defined in the domain f .D/, is a quasiconformal map with the same complex dilatation as that of f . The proof follows from the composition formula and the Weil lemma (cf. [1]). 1.2.2 The Existence Theorem Theorem 4 (Existence Theorem). For any measurable bounded function  in the complex plane C with kk1 < 1 there exists a solution w of the Beltrami equation N D @w @w which is a quasiconformal map with complex dilatation equal to  almost everywhere. We shall describe the main steps of the proof of this theorem while the details can be found in the Ahlfors’ book [1]. Suppose first that the function  has a compact support. Lemma 1. Under the assumption that  has compact support there exists a unique solution w of the Beltrami equation (1.2), satisfying the conditions w.0/ D 0;

@w  1 2 Lp

where p > 2 is some number which has to be chosen later. Such a solution of the Beltrami equation will be called normal. Consider instead of the Beltrami equation (1.2) the system of equations @w D f; N D f @w

(1.3)

where f is some measurable bounded function. It is clear that any solution of the system (1.3) yields a solution of Beltrami equation (1.2). However, the system (1.3)

1.2 Lecture II. Uniqueness and existence theorems

7

is overdetermined – it is solvable only if the mixed derivatives of the right-hand sides coincide almost everywhere: N D @.f / D f @ C @f: @f

(1.4)

In order to find a function f giving a solution to the system (1.3), and hence for Beltrami equation (1.2), we shall proceed in the following way. First, using the 2nd N equation in (1.3), we write w in the form of the solution of the @-equation, given by the Cauchy–Green formula, plus an unknown holomorphic function W . Next, by substituting this expression into the 1st equation in (1.3), we shall obtain an integral equation for the function f . In this equation we can fix the function W by the condition @w  1 2 Lp . The integral operator in this equation is a contraction due to the estimate kk1 < 1. So the obtained integral equation could be solved by iteration. This is the general strategy of solution of the system (1.3). Before we start to realize it, let us recall some necessary properties of the Cauchy– Green integral operator, given by the formula Z   1 1 1 P h.z/ D  h./ d d ;  D  C i :   C z  ThisRoperator differs from the standard Cauchy–Green operator, given by the integral  1 C h./ d d , only by the normalization condition P h.0/ D 0. The Cauchy– z Green operator P h.z/ is correctly defined on functions h 2 Lp with any p > 2 and determines a continuous (even Hölder-continuous of order 1  p2 ) function in z. The partial derivatives of the function P h.z/ in the generalized sense satisfy the equations N h/ D h; @.P

@.P h/ D T h

(1.5)

where T is an integral operator of Calderon–Zygmund type: Z 1 h./ T h.z/ D  P.V. d d : 2  C .  z/ The integral in this formula should be taken in the principal value sense, in other words, it is equal by definition to Z 1 h./ T h.z/ D  lim d d :  !0 jzj> .  z/2 The Calderon–Zygmund operator is correctly defined on functions h 2 Lp with any p > 1 and bounded in the Lp -norm: kT hkLp  Cp khkLp : Moreover, the bound Cp can be chosen in such a way that Cp ! 1 for p ! 2. Using this property, we can choose the number p > 2 so that the inequality kk1 Cp < 1 is satisfied.

8

1 Quasiconformal maps

Return to the scheme of the construction of solution of system (1.3), proposed before. Consider the function N W WD w  P .@w/: The partial derivative of this function in zN vanishes which means that W is an entire function. On the other hand, the condition @w  1 2 Lp implies that the derivative of this function in z, equal to N @W D @w  T .@w/; N D @w 2 Lp (recall that  by satisfies the condition @W  1 2 Lp since @w assumption has a compact support!). This is possible only if @W  1 H) W .z/ D z C const: The constant in the last equality is equal to zero because of the normalization w.0/ D 0, hence N w D z C P .@w/: By differentiating this relation in z, we obtain N D 1 C T .@w/ D f @w D 1 C T .@w/

(1.6)

due to the 1st equation in (1.3). Thus, we get an integral equation on the function @w in which the operator T B  is a contraction since kT B kLp  kk1 Cp < 1: Substituting f D h C 1 into the equation (1.6), we obtain an integral equation on the function h: h D T .h/ C T: The unique solution h 2 Lp of this equation yields the required solution of Beltrami equation (1.2), given by the formula  D P ..h C 1// C z: Indeed, .h C 1/ 2 Lp since  has compact support. So the function P ..h C 1// is correctly defined and continuous. Moreover, w.0/ D 0 and N D @P N ..h C 1// D .h C 1/; @w @w D @P ..h C 1// C 1 D T ..h C 1// C 1 D h C 1: Lemma 2. Suppose that in addition to the assumptions of Lemma 1 the function  is C 1 -smooth. Then under this condition the normal solution of system (1.3) is a C 1 -smooth homeomorphism.

1.2 Lecture II. Uniqueness and existence theorems

9

Remark 3. For the validity of Lemma 2 it is sufficient to require that the function  has a generalized derivative @ 2 Lp (cf. [1]). Proof. Recall that for the solvability of the system @w D f;

N D f @w

it is necessary that the following relation is true: N D f @ C @f: @f Suppose that we have managed to construct a solution w of system (1.3) which is a C 1 -smooth homeomorphism. Then its Jacobian, equal to N 2 D .1  jj2 /jf j2 ; j@wj2  j@wj should be strictly positive, in other words, the function f should have no zeros. So it is reasonable to look for a function having the form f D eg : For the function g the relation (1.4) will transform into the equation N D @g C @: @g The solution of this equation can be reduced to the solution of the overdetermined system of equations @g D h; N D h C @ @g where h is some measurable function. We write the function g in the form g D P .h/ C P .@/ C const: Then for the function h we shall obtain the equation @g D T .h/ C T .@/ D h: This equation has a unique solution h 2 Lp . Hence the introduced function g D P .h/ C P .@/ C const is correctly defined and continuous everywhere, including the point z D 1 (recall that  has compact support!). We choose now the constant in the formula for g so that g.z/ ! 0

for z ! 1:

The function f D e g satisfies the relation (1.4). Now, as in the proof of Lemma 1, one can show that system (1.3) has a solution which is C 1 -smooth in this case. If we

10

1 Quasiconformal maps

normalize it by the condition w.0/ D 0 then this solution will coincide with the normal one since for z ! 1 we have f .z/ ! 1 H) @w ! 1: The Jacobian of the constructed map w coincides with N 2 D .1  jj2 /e 2g > 0; j@wj2  j@wj i.e. the map w is locally one-to-one. Since w.z/ ! 1 for z ! 1 (because @w ! 1), the map w is a global homeomorphism.  The condition  2 C 1 may be removed using the approximation of the original function  by C 1 -smooth functions (cf. [1]). Lemma 3. The condition of compactness of the support of  can be removed. Before we switch to the proof of this assertion, let us introduce the following notation. In order to determine uniquely a solution of Beltrami equation in the extended x which is a priori defined only up to fractional-linear maps, it is complex plane C, sufficient to fix any three of its values. We shall require that the considered solution fixes three points 0, 1, 1. Such a solution of equation (1.2) will be denoted by w  . Proof. Switching to the proof of lemma, consider first the case in which   0 in a neighborhood of the point z D 0. Introduce the reflection Q of the function  with respect to the unit circle, defined by the formula  

.z/ Q D

1 z2 : z zN 2

This function has a compact support in C, hence the Beltrami equation with Beltrami differential Q admits a solution w. z Then the original Beltrami equation (1.2) with differential  will have the solution w.z/ D Indeed,

w z

1 1 : z

 

1 1 1 @w.z/ D  2  2  @w z ; 1 z z w z z

 

1 1 1 N z @w.z/ D  2  2  @N w 1 zN z w z z

N D @w. which implies that @w In the general case we represent  in the form  D 1 C 2

1.2 Lecture II. Uniqueness and existence theorems

11

where 1  0 in a neighborhood of 1 and 2  0 in a neighborhood of 0. Of course, we cannot conclude from this representation that w  D w 1 B w 2 due to the nonlinear character of the composition formula. However, this representation implies that w  D w  B w 2 () w  D w  B .w 2 /1 ; where



D





  2 @w 2 B .w 2 /1 ; N 2 1  N 2 @w

and such a function has compact support.



Consequently, we have proved the existence theorem for quasiconformal maps in the following formulation. x there Theorem 5 (Existence Theorem). For any measurable bounded function  in C exists a unique quasiconformal map w  with the complex dilatation equal to  almost everywhere which leaves the points 0, 1, 1 invariant. This theorem implies the following generalization of the Riemann Mapping Theorem also called the Measurable Riemann Mapping Theorem. x with non-trivial Theorem 6. Let D and D 0 be two simply connected domains in C boundaries (i.e. with boundaries consisting of more than one point). If  2 L1 .D/ with kk1 < 1 then there exists a quasiconformal map w W D ! D 0 with the complex dilatation equal to  almost everywhere. We leave the proof of this assertion as an exercise. Brief content of Lecture II Quasiconformal homeomorphisms D ! D form a group with respect to the composition. There is a formula expressing the Beltrami differential of the composition of two quasiconformal maps via their Beltrami differentials. Uniqueness Theorem: f D g H) the maps f B g 1 and g B f 1 are conformal. Existence Theorem: for any  2 L1 .C/ with kk1 < 1 there exists a solution N D  @w which is a quasiconformal map with complex of Beltrami equation: @w dilatation . These two theorems imply that there exists a unique normalized solution w  of x leaving the points 0, 1, 1 the Beltrami equation in the extended complex plane C, invariant.

12

1 Quasiconformal maps

1.3 Lecture III. Quasisymmetric homeomorphisms 1.3.1 Boundary behavior of quasiconformal maps Theorem 7 (Mori theorem). Let w W  !  be a K-quasiconformal homeomorphism of the unit disk  D fz 2 C W jzj < 1g onto itself, normalized by the condition w.0/ D 0. Then for any z1 ¤ z2 from  the following estimate holds true: jw.z1 /  w.z2 /j < 16jz1  z2 j1=K : In other words, the homeomorphism w is Hölder continuous in  with Hölder exponent 1=K. The proof of this theorem which we omit could be found in the book [1], Ch. III, Sec. C. It immediately implies the following result. Corollary 1. Any quasiconformal homeomorphism  !  of the unit disk  onto x ! . x More generitself continuously extends to a homeomorphism of the closures  ally, any quasiconformal homeomorphism of a Jordan domain D onto another Jordan x !D x 0. domain D 0 extends continuously to a homeomorphism of the closures D For quasiconformal maps we have a theorem on normal convergence which is analogous to the corresponding theorem for holomorphic maps. Theorem 8 (Normal convergence theorem). Suppose we are given a sequence of Kquasiconformal maps wn in a domain D which is converging uniformly on compact subsets to a map w. Then the limiting function w is either a constant or a K-quasiconformal map. The proof of this theorem can be found in the book [16], Ch. 1, Sec. 2.3. 1.3.2 Conformal rectangles. Geometric definition of quasiconformal maps. Recall first the notion of the conformal modulus which is necessary for the geometric definition of quasiconformal maps. It is well known that for any triples .z1 ; z2 ; z3 / and .w1 ; w2 ; w3 / of pairwise disx there exists joint points z1 , z2 , z3 and w1 , w2 , w3 in the extended complex plane C x a conformal (hence, fractional-linear) transform of C onto itself sending the points .z1 ; z2 ; z3 / to .w1 ; w2 ; w3 /. An analogous assertion for quadruples of pairwise disjoint points .z1 ; z2 ; z3 ; z4 / and .w1 ; w2 ; w3 ; w4 / is already not true (except in certain special cases). We define conformal rectangle Q.z1 ; z2 ; z3 ; z4 / to be a Jordan domain in the exx such that its boundary contains the points z1 , z2 , z3 , z4 and tended complex plane C the consecutive motion z1 7! z2 7! z3 7! z4 determines the positive direction at the boundary of the domain (for which the domain remains to the left side while moving

1.3 Lecture III. Quasisymmetric homeomorphisms

13

along the boundary). Any such conformal rectangle may be conformally mapped onto a Euclidean rectangle ….w1 ; w2 ; w3 ; w4 / and such a transform is uniquely determined up to similarity. (Prove this fact!) Now suppose that this conformal transform maps the given conformal rectangle Q.z1 ; z2 ; z3 ; z4 / onto the Euclidean rectangle ….0; a; a C i b; i b/. In this case we call the number MQ.z1 ; z2 ; z3 ; z4 / WD a=b the conformal modulus of the rectangle Q.z1 ; z2 ; z3 ; z4 /. It is evident that this number is a conformal invariant and the equality MQ.z1 ; z2 ; z3 ; z4 / D MQ.w1 ; w2 ; w3 ; w4 / is necessary and sufficient for the conformal equivalence of conformal rectangles Q.z1 ; z2 ; z3 ; z4 / and Q.w1 ; w2 ; w3 ; w4 /. Moreover, the following result, proved in [17], is true. Theorem 9. Let w W D ! D 0 be an orientation-preserving homeomorphism. If w leaves the moduli of conformal rectangles invariant then w is conformal. Consider for a given domain D  C the set fQg of all conformal rectangles Q.z1 ; z2 ; z3 ; z4 / having their closures inside D. Theorem 10. Let w W D ! D 0 be an orientation-preserving homeomorphism for which the quantity K WD sup fQg

MQ .w.z1 /; w.z2 /; w.z3 /; w.z4 // ; MQ.z1 ; z2 ; z3 ; z4 /

called the maximal dilatation, is finite. Then the map w is K-quasiconformal. In fact, the property, formulated in this theorem, could be taken for the definition of K-quasiconformal maps. The proof of the equivalence of this definition to the original one cf. in [17]. Corollary 2. Any 1-quasiconformal map is necessarily conformal. 1.3.3 Quasisymmetric homeomorphisms. As it was pointed out in Section 1.3.1, quasiconformal homeomorphisms of Jordan domains automatically extend to homeomorphisms of their closures. One can ask when the converse assertion is true, i.e. when a given homeomorphism of the boundaries of two domains can be extended to a quasiconformal homeomorphism of the domains themselves? Let us consider this problem first in the case when both domains coincide with the upper halfplane H D fz 2 C W Im z > 0g. The boundary value of a quasiconformal homeomorphism H ! H is an orientation-preserving homeomorphism R ! R. In other words, it is given by a monotonically increasing homeomorphism f W R ! R

14

1 Quasiconformal maps

with the condition f .x/ ! 1 for x ! 1. We call the maximal dilatation of the homeomorphism f the quantity MH .f .x1 /; f .x2 /; f .x3 /; f .x4 // MH.x1 ; x2 ; x3 ; x4 / fxi g

K D sup

(1.7)

x such that where the supremum is taken over all quadruples of points x1 ; x2 ; x3 ; x4 in R the consecutive motion x1 7! x2 7! x3 7! x4 determines the positive direction on the real axis R (we denote by H.x1 ; x2 ; x3 ; x4 / the conformal rectangle equal to the upper halfplane with fixed points .x1 ; x2 ; x3 ; x4 /). From the geometric definition of K-quasiconformal maps, given above (cf. Theorem 10) it follows that the maximal dilatation of a homeomorphism f W R ! R, which is the boundary value of a K-quasiconformal homeomorphism w W H ! H, coincides with K. Let us choose the points x1 D x  t; x2 D x; x3 D x C t; x4 D 1

with t > 0

in the formula (1.7). For them the quantity MH.x  t; x; x C t; 1/ D 1, and it follows from the finiteness of the maximal dilatation that 1 f .x C t /  f .x/   C.K/: C.K/ f .x/  f .x  t / Problem 3. Prove this assertion. Definition 5. A monotonically increasing homeomorphism f W R ! R is called M quasisymmetric if there exists M  1 such that f .x C t /  f .x/ 1   M; M f .x/  f .x  t /

x 2 R;

(1.8)

for all t > 0. Hence, the boundary value of a K-quasiconformal homeomorphism H ! H is an M -quasisymmetric homeomorphism R ! R with M D C.K/. The converse assertion is also true. Theorem 11 (Beurling–Ahlfors theorem). Let f W R ! R be an M -quasisymmetric homeomorphism. Then there exists a quasiconformal homeomorphism w W H ! H such that its boundary value coincides with f and maximal dilatation is bounded by a number K.M / which can be chosen so that K.M / ! 1 for M ! 1. Idea of the proof (confer the complete proof in the book [1], Ch. IV, Sec. B). Define x by the formula a map w, which is defined on the closure H, Z Z i 1 1 1 Œf .x C ty/ C f .x  ty/ dt C Œf .x C ty/  f .x  ty/ dt: w.xCiy/ D 2 0 2 0

1.3 Lecture III. Quasisymmetric homeomorphisms

15

It is evident that w.x/ D f .x/ on the real line. If we introduce the notation Z 1 Z 1 xCy ˛.x; y/ D f .x C ty/dt D f .t /dt; y x 0 Z 1 Z 1 x ˇ.x; y/ D f .x  ty/dt D f .t /dt y xy 0 then the formula for w will be rewritten in the form w.x C iy/ D

˛Cˇ ˛ˇ Ci : 2 2

The geometric meaning of the functions ˛ and ˇ is clear: the function ˛.x; y/ assigns to a point x C iy 2 H the mean value of the function f on the interval Œx; x C y, while the function ˇ.x; y/ assigns to this point the mean value of f on the interval Œx  y; x. The functions ˛ and ˇ are continuously differentiable with respect to x and y and the map w W H ! H, determined by these functions, has a positive Jacobian (a precise value of this Jacobian is computed in the book [1]). It implies, together with the fact that the boundary value of w equal to f is a monotonically increasing homeomorphism R ! R, that w determines a homeomorphism H ! H. The quasiconformality condition of w is rewritten in terms of an inequality, relating the values of the function f in the points x, x C y and x  y, which is provided by the condition (1.8). In the case of the unit disk  it is convenient to formulate the quasisymmetricity condition of Berling–Ahlfors (1.8) in terms of the cross ratios. Recall that the cross (or double) ratio of four pairwise disjoint points z1 , z2 , z3 , z4 on the complex plane is the quantity CR.z1 ; z2 ; z3 ; z4 / WD

z4  z1 z3  z1 W : z4  z2 z3  z2

The equality of cross ratios CR.z1 ; z2 ; z3 ; z4 / D CR.w1 ; w2 ; w3 ; w4 / is necessary and sufficient for the existence of a conformal transform of the extended complex plane, mapping the points .z1 ; z2 ; z3 ; z4 / to the points .w1 ; w2 ; w3 ; w4 /. Definition 6. An orientation-preserving homeomorphism f of the unit circle S 1 onto itself is called quasisymmetric if for some 0 < < 1 it satisfies the condition 1 1 .1  /  CR .f .z1 /; f .z2 /; f .z3 /; f .z4 //  .1 C / 2 2

(1.9)

for any four-tuple of pairwise disjoint points z1 , z2 , z3 , z4 on S 1 with the cross ratio CR.z1 ; z2 ; z3 ; z4 / D 12 .

16

1 Quasiconformal maps

The condition (1.9) is an analogue of the Beurling–Ahlfors condition (1.8) for the unit disk . This condition, as in the case of the upper halfplane H, guarantees the quasiconformal extendability of a quasisymmetric homeomorphism f W S 1 ! S 1 inside . Theorem 12. Suppose that f is an orientation-preserving homeomorphism of the unit circle S 1 onto itself which satisfies the condition (1.9). Then it can be extended to a quasiconformal homeomorphism w W  ! . Remark 4. Note that the quasiconformal extension of a quasisymmetric homeomorphism, constructed by Beurling–Ahlfors, is not at all unique. For example, Douady and Earle (cf. [10]) have found an explicit extension operator E assigning to a quasisymmetric homeomorphism f its extension to a quasiconformal homeomorphism w of  which is conformally invariant in the sense that g.w B f / D w B g.f / for any fractional-linear automorphism g of . Problem 4. Prove that any orientation-preserving diffeomorphism of the circle S 1 onto itself is quasisymmentric, i.e. it can be extended to a quasiconformal diffeomorphism of the unit disk . Brief content of Lecture III Boundary behavior: Quasiconformal homeomorphisms D ! D 0 of Jordan domains x !D x 0. D; D 0 extend continuously to homeomorphisms of the closures D Geometric definition of quasiconformal maps: a homeomorphism w W D ! D 0 , preserving the orientation, is K-quasiconformal () it has a finite maximal dilatation K WD sup fQg

MQ .w.z1 /; w.z2 /; w.z3 /; w.z4 // MQ.z1 ; z2 ; z3 ; z4 /

x taken over all conformal rectangles Q contained in D along with their closures Q. Quasisymmetric homeomorphisms of the upper halfplane: a monotonically increasing homeomorphism f W R ! R is called quasisymmetric if for some M  1 the Beurling–Ahlfors condition 1 f .x C t /  f .x/  M M f .x/  f .x  t / is satisfied for all x 2 R, t > 0. Beurling–Ahlfors theorem: a monotonically increasing homeomorphism f W R ! R is quasisymmetric () it can be extended to a quasiconformal homeomorphism w W H ! H of the upper halfplane H. In the case of the unit disk the quasisymmetricity condition of an orientationpreserving homeomorphism f W S 1 ! S 1 is formulated in terms of cross ratios: a

1.3 Lecture III. Quasisymmetric homeomorphisms

17

homeomorphism f W S 1 ! S 1 is quasisymmetric if for some , 0 < < 1, the following condition 1 1 .1  /  CR .f .z1 /; f .z2 /; f .z3 /; f .z4 //  .1 C / 2 2 is satisfied for all z1 ; z2 ; z3 ; z4 2 S 1 with cross ratio CR.z1 ; z2 ; z3 ; z4 / D 12 . The Beurling–Ahlfors theorem is also true in the case of the unit disk : if f W S 1 ! 1 S is an orientation-preserving diffeomorphism then it is quasisymmetric, i.e. it can x be extended to a diffeomorphism of the closed unit disk .

2 Universal Teichmüller space

In this chapter we introduce the universal Teichmüller space T and present basic facts about this space. We give three different definitions of T . The first definition in terms of quasisymmetric homeomorphisms is given in Section 2.1.1, the second one – in terms of quasidisks – in Section 2.1.2 and the third one – in terms of Beltrami differentials – also in Section 2.1.2. Basic metric and topological properties of the universal Teichmüller space are collected in Section 2.2.1. The Bers embedding of universal Teichmüller space T into the space of holomorphic quadratic differentials in the disk is constructed in Section 2.2.3. With the help of this embedding, we introduce a natural complex structure on T . This construction uses the properties of the Schwarzian derivative, communicated in Section 2.2.2. In Section 2.2.5 we address the problem of the existence of a Kähler metric on the space T and define a densely defined Kähler quasimetric on T . In the same section we introduce quasisymmetric vector fields on the circle associated to functions from the Zigmund space.

2.1 Lecture IV. Definition of the universal Teichmüller space 2.1.1 Definition in terms of quasisymmetric homeomorphisms. The Composition Theorem from Section 1.2.1 implies that quasiconformal homeomorphisms of the unit disk  form a group with respect to the composition operation. Hence quasisymmetric homeomorphisms of the unit circle S 1 also form a group with respect to this operation. We denote it by QS.S 1 /. According to Problem 4 at the end of Section 1.3.3, the group QS.S 1 / contains the group DiffC .S 1 / of orientation-preserving diffeomorphisms of S 1 . So we have the following chain of embeddings: Möb.S 1 /  DiffC .S 1 /  QS.S 1 /  HomeoC .S 1 /: In this chain, HomeoC .S 1 / denotes the group of orientation-preserving homeomorphisms of S 1 and Möb.S 1 / is the group of fractional-linear automorphisms of the unit disk , restricted to S 1 . Definition 7. The space T D QS.S 1 /=Möb.S 1 / is called the universal Teichmüller space.

20

2 Universal Teichmüller space

The space T may be identified with the subspace of QS.S 1 /, consisting of normalized quasisymmetric homeomorphisms of the circle, fixing three points on the circle S 1 . Usually we take for such points ˙1 and i . The universal Teichmüller space T contains as a subspace the space  D DiffC .S 1 /=Möb.S 1 / which can be identified with the space of normalized quasisymmetric diffeomorphisms of the circle. This space is studied in detail in Chapter 4. 2.1.2 Definition in terms of Beltrami differentials. Since the notion of quasiconformality may be defined in terms of Beltrami differentials (cf. Section 1.1.3), it is convenient to have a definition of the universal Teichmüller space T directly in terms of these differentials. Denote the space of Beltrami differentials in the unit disk  by B./  B.1;1/ ./: According to Section 1.1.3, it can be identified with the unit ball in the complex Banach space L1 ./. Let  2 B./ be a Beltrami differential in the disk . We extend it to a Beltrami x using the reflection with respect to the differential O in the extended complex plane C, circle S 1 , by setting   z2 1 WD .z/ 2 for z 2 : O z zN Applying the Existence Theorem for quasiconformal maps (Theorem 4 from Section 1.2.1) to the Beltrami equation with the extended Beltrami differential , O we can find a normalized quasiconformal homeomorphism w of the extended complex x with the complex dilatation . plane C O According to the Uniqueness Theorem (Theorem 3 from Section 1.2.1), this homeomorphism w should be symmetric with respect to S 1 , i.e. w .1=z/ D 1=w .z/ (otherwise, the reflected solution will give another normalized solution with the same dilatation in the disk). Hence, w should map the circle S 1 into itself. In this way, we can assign to the original Beltrami differential  the normalized quasisymmetric homeomorphism of the circle w : B./ 3  7! w jS 1 2 T : This map is bijective modulo the following equivalence relation of Beltrami differentials:   () w jS 1  w jS 1 : Hence, the universal Teichmüller space T may be identified with the quotient T D B./=  :

2.1 Lecture IV. Definition of the universal Teichmüller space

21

There is another natural method of continuation of a given Beltrami differential x by setting  2 B./ to a Beltrami differential L in the extended complex plane C x n : x .z/ L  0 for z 2  WD C Applying the Existence Theorem for quasiconformal maps to the Beltrami equation with the extended Beltrami differential , L we obtain a quasiconformal homeomorphism x with complex dilatation . w  of the extended complex plane C L It is convenient to normalize it by fixing the points 0, 1, 1. The constructed normalized quasiconformal homeomorphism w  will be conformal in the complement  to the closed unit x disk . We call the image of the unit disk  (resp. of the unit circle S 1 ) under a quasiconformal map a quasidisk (resp. quasicircle). In terms of this definition we have just constructed a map x B./ 3  7! quasidisk  WD w  ./ in C: This map is bijective modulo the following equivalence relation of Beltrami differentials:  () w  j   w j  : Lemma 4. The introduced equivalence relations of Beltrami differentials coincide, i.e.   ()  : Proof. Suppose first that  , i.e. w   w in  . Consider the maps 1 W  !  w  B w

and w B w 1 W  !  :

Note that  D  , since w  D w on S 1 . Both maps are conformal. For example, 1 w  Bw is conformal in the disk , since w  and w have the same complex dilatation  in . An analogous argument can be applied to the map w B w 1 . Note also that the introduced maps send the three given points ˙1, i on S 1 to the three points w  .˙1/ D w .˙1/, w  .i / D w .i / on @ . It follows that these maps coincide, i.e. 1 w  B w  w B w 1 on ; hence, on S 1 . But w  D w on S 1 which implies that w D w on S 1 , i.e.   . In the opposite direction, let   , i.e. w D w on S 1 . Consider the map ´ w  B .w /1 on w . /; wD 1 w  B w B w B .w /1 on w ./. Note that the boundary values of both maps, defined respectively on w . / and w ./, coincide on the quasicircle w .S 1 /, since w D w on S 1 . The introduced

22

2 Universal Teichmüller space

homeomorphism w is conformal in the quasidisk w . /, since w  and w are con1 formal on  . It is conformal in the quasidisk w ./, since the maps w  B w and 1  w B .w / are conformal (because the complex dilatations of w and w , as well as of w and w , coincide in the unit disk ). It follows that the homeomorphism x according to the following w is conformal in the whole extended complex plane C, assertion the proof of which is left as an exercise. x which is KProblem 5. Let w be a homeomorphism of the extended complex plane C, quasiconformal in the complement of the quasicircle S. Then w is K-quasiconformal x everywhere in C. The formulated problem implies that the introduced homeomorphism w is conx i.e. w coincides with a fractional-linear transform. This formal everywhere in C, transform fixes the points 0, 1, 1 because of the normalization, which implies that w  id. Hence, w  D w on  , i.e.  .  Thanks to the proved lemma, we obtain two more interpretations of the universal Teichmüller space: x T D fthe space of normalized quasidisks in Cg D fthe space of normalized quasiconformal maps x ! C, x which are conformal in the disk  g: C

(2.1)

The relation between the two realizations of the universal Teichmüller space as the space of normalized quasisymmetric homeomorphisms of the circle S 1 and the space x may be established in a more explicit way. of normalized quasidisks in C To see this we introduce the following welding problem, which is of independent interest. Welding problem. Let f be a homeomorphism of the circle S 1 onto itself. We want to find conformal maps wC and w , defined respectively in the disk  and its complement  , such that 1 B w on S 1 : (2.2) f D wC The pair .wC ; w / is normalized if the maps w˙ fix the points ˙1; i . Lemma 5 (Welding lemma). Let f be a normalized quasisymmetric homeomorphism of the circle S 1 onto itself. Then the welding problem (2.2) admits a unique normalized solution. Proof. By the Beurling–Ahlfors theorem, for a given quasisymmetric homeomorphism f there exists a normalized quasiconformal homeomorphism w W  !  such that x by zero outside  wjS 1 D f . Denote its complex dilatation by  and extend  to C

2.1 Lecture IV. Definition of the universal Teichmüller space

23

x with to a Beltrami differential . L Let w z  be a quasiconformal homeomorphism of C complex dilatation , L which fixes the points ˙1; i 2 S 1 . Then the maps 1 z ; wC W D w z  j B w  W  ! quasidisk  z w W D w z  j  W  ! quasidisk  

(2.3) (2.4)

are conformal respectively in  and  (conformality of wC follows from the fact that homeomorphisms w z  and w have the same complex dilatation  in the disk ). Hence they determine a normalized solution of the welding problem (2.2). If .vC ; v / is another solution of this problem then we shall have on S 1 the following equalities: 1 v D vC B f D vC B w D vC B .wC B w /: Consider the map

´ vD

vC B w v

on  [ S 1 ; on  :

x which is quasiconformal everywhere outside S 1 . AcIt is a homeomorphism of C cording to Problem 5 from Section 2.1.2, this homeomorphism v is quasiconformal x Since vC is conformal in the disk , v has the same complex everywhere in C. z  , which implies that dilatation , as w z  , and fixes the points ˙1, i . Hence, v D w 1 vC D w z  B w D wC ;

v D w z  D w :



Let us return to the correspondence, mentioned above: fnormalized quasisymmetric homeomorphisms of S 1 g x ! fnormalized quasidisks in Cg:

(2.5)

If f is a normalized quasisymmetric homeomorphism S 1 ! S 1 then it admits a unique normalized welding of the form 1 f D wC B w 1 where wC D w z  j B w  , w D w z  j  and  is the complex dilatation of the normalized quasiconformal homeomorphism w, associated with f . We assign the normalized quasidisk  D w  ./ to this homeomorphism f . Conversely, if  is a normalized quasidisk, associated with the quasiconformal map w  with complex dilatation , then we consider the maps 1 wC D w z  B w on 

and w D w z  on  :

These maps are conformal and fix the points ˙1; i on S 1 . We assign to these maps the quasisymmetric homeomorphism of the circle S 1 onto itself, given by the formula 1 f D wC B w

on S 1 :

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2 Universal Teichmüller space

This is precisely the normalized quasisymmetric homeomorphism S 1 ! S 1 we were looking for. Brief content of Lecture IV First definition of universal Teichmüller space: T D QS.S 1 /=Möb.S 1 /: Second definition of universal Teichmüller space: We can extend a given Beltrami x either by reflection with respect differential  2 B./ to the extended complex plane C 1 to the circle S , or by zero outside the disk . Solving the Beltrami equation for the extended Beltrami differentials, we obtain two normalized quasiconformal homeomorx The first extension determines the quasiconformal homeomorphism w , phisms on C. preserving the disk , the second is the quasiconformal homeomorphism w  which x Using the latter homeomorphism w  , we can is conformal in the complement of . x or define the universal Teichmüller space as the space of quasidisks  D w  ./ in C x as the space of normalized quasiconformal maps of C, conformal in the complement x of . Third definition of universal Teichmüller space: T D B./=  ; i.e. T is the quotient of the space of Beltrami differentials in the disk  modulo the following equivalence relation:   () w D w on S 1 () w  D w in the x complement of .

2.2 Lecture V. Properties of the universal Teichmüller space 2.2.1 Metric and topological properties of T . We introduce the Teichmüller distance between the points of the space T . If f and g are two normalized quasisymmetric homeomorphisms of S 1 then the distance between them is given by the quantity dist.f; g/ D

1 log KŒg B f 1  2

where KŒg B f 1  is the maximal dilatation of the quasisymmetric homeomorphism g B f 1 which is defined in the same way as the maximal dilatation of a monotonically increasing homeomorphism f W R ! R from (1.7). In terms of complex dilatations this quantity is equal to  k1 1 C k 1 1  N dist.Œ; Œ / D min log  2 1  k 1 k1 N

25

2.2 Lecture V. Properties of the universal Teichmüller space

where the minimum is taken over all  2 Œ, 2 Œ , and Œ denotes the class of Beltrami differentials, containing , in the quotient T D B./= . The expression on the right-hand side of the last definition is reminiscent of the formula for the hyperbolic distance in the disk . Namely, if we define the distance between the functions  and by the formula k  k WD ess supz2 r ..z/; .z// ; where r is the hyperbolic distance in the disk , then dist.Œ; Œ / D min k  k

where the minimum is taken over all  2 Œ; 2 Œ . We list now the main topological properties of the space T , their proofs may be found in the book [16], Ch. III, Sec. 2. Assertion 1. The space T is linearly connected. Assertion 2. The space T is complete, i.e. any Cauchy sequence in T converges. Assertion 3. The space T is contractible. Assertion 4. The space T is not a topological group. In other words, the operation of composition of normalized quasisymmetric homeomorphisms S 1 ! S 1 is not continuous in the Teichmüller metric. 2.2.2 Schwarzian derivative. To define a complex structure on the space T , we need to recall the main properties of Schwarzian derivative. The Schwarzian derivative of a conformal map f is equal to the quantity 

S Œf  D

f 00 f0

0





1 f 00 2 f0

2



1 f 000 3 f 00 D .log f 0 /0  .log f 0 /2 D 0  2 f 2 f0

2

:

Here are the properties of Schwarzian derivative to be used later (try to prove these properties by yourself). Property 1. If f is a fractional-linear map then SŒf  D 0: Property 2. If the function f has no zeros in the domain of definition then 



1 SŒf  D S : f This property may be used for the definition of the Schwarzian derivative of a locally injective meromorphic map.

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2 Universal Teichmüller space

Property 3. The Schwarzian derivative satisfies the composition formula: SŒf B g D .SŒf  B g/ g 02 C S Œg: In particular, if g is a Möbius transform then SŒf B g D .SŒf  B g/ g 02 : In the case when f is a Möbius transform SŒf B g D S Œg; i.e. S Œf  is invariant under the action of Möbius transforms from the left: SŒf  D S



af C b cf C d



for

az C b x 2 Möb.C/: cz C d

The composition formula may be used for the definition of the value of S Œf  at the x Namely, according to this formula this value should be equal to point 1 2 C. SŒf .1/ D lim z 4 S Œ'.z/ z!0

where '.z/ WD f

1 z

.

Property 4. This property motivates the usage of the term “derivative” with respect to S Œf . Namely, for any z0 there exists a unique Möbius transform h, for which the following limit .h B f /.z/  z 1 lim D S Œf .z0 / 3 z!z0 .z  z0 / 6 exists. Property 5. Let ' be a holomorphic function in a simply connected domain D. Then there exists a meromorphic function f in D such that SŒf  D ': This function f is determined uniquely by ' up to Möbius transforms. (Cf. [16], Ch. II, Sec. 1.2.) If a domain D is conformally equivalent to the disk  then it is possible to define the norm of S Œf , using the density  of the Poincaré metric in the disk . Namely, kSŒf kD WD sup

z2D

jS Œf .z/j : .z/2

The norm of Schwarzian derivative kSŒf kD measures the distance from a given conformal map to the set of fractional-linear maps.

2.2 Lecture V. Properties of the universal Teichmüller space

27

Property 6. If f is a conformal map, given in the unit disk , then the following sharp estimate kSŒf k  6 is true. The proof of this property, which may be found in the book [16], Ch. II, Sec. 1.6, is based on the following well-known Area Theorem. Theorem 13 (Area Theorem). Let f be an univalent meromorphic function in the x which has the following power decomposition in a neighborhood complement  of  of infinity 1 X bn : f .z/ D z C zn nD0 Then

1 X

njbn j2  1;

nD1

moreover this estimate is sharp. The proof of this theorem may be found in the book [16]), but first try to prove it by yourself. The Area Theorem implies, in particular, that jb1 j  1. The equality in this estimate i is attained only on the function f .z/ D z C ez . 2.2.3 Bers embedding. We construct an embedding of the universal Teichmüller space T into the space of holomorphic quadratic differentials in the disk. We associate with a point Œ 2 T the normalized quasiconformal homeomorx so we can consider its phism w  . Then w  is conformal in the complement  of  Schwarzian derivative S Œw  j   : The obtained function in  does not depend on the choice of  2 Œ, since by Lemma 4 the conformal map w  j  does not depend on this choice. Moreover, this function is holomorphic in z 2  and transforms under conformal changes of variable as a quadratic differential due to the Property 3 of the Schwarzian derivative. The map Œ 7! S Œw  j   is an embedding, since the equality S Œw  j   D S Œw j   implies, according to the Property 5 of the Schwarzian derivative, that w  j  D w j  ; i.e.   .

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2 Universal Teichmüller space

So we have constructed an embedding ‰ W T ! Q. / of the universal Teichmüller space T into the space Q. /  B.2;0/ . / of holomorphic quadratic differentials in the disk  . This map is called the Bers embedding. Note that Q. / is a complex Banach space, provided with the natural hyperbolic norm: Q. / D f

D

.z/dz 2 W k kQ WD sup .1  jzj2 /2 j .z/j < 1g: z2 

It may be shown (cf. [16], Ch. III, Sec. 4.3) that the embedding ‰ is a homeomorphism of the space T onto its image in Q. /. The description of the image ‰.T / in Q. / is a difficult problem, however the following theorem is true. Theorem 14. The image ‰.T / in Q. / is an open connected contractible subset in Q. / which contains the open ball B.0; 2/ of radius 2 with center at the origin and is contained in the ball B.0; 6/. The proof of the facts given in this theorem may be found in the book [16], Ch. III, Secs. 3, 4. 2.2.4 Complex structure of the space T . We introduce the complex structure on the space T by setting it equal to the complex structure induced from the complex Banach space Q. / via the Bers embedding. Otherwise, the complex structure on T may be induced from the complex Banach space B./  L1 ./ by the natural projection B./ ! T D B./=  : It turns out that both methods give the same result. More precisely, the following theorem is true. Theorem 15. The composition of the natural projection B./ ! T with the Bers embedding, yielding the map F W B./ ! Q. /; is a holomorphic map of complex Banach spaces. This theorem is proved in the book [18], Sec. 3.4. We give an explicit description of the tangent map dF at the origin, i.e. at the point   0. This point is sent by the map F to the class Œid D ŒMöb.S 1 / of the quotient T D QS.S 1 /=Möb.S 1 /:

2.2 Lecture V. Properties of the universal Teichmüller space

29

(A formula for the differential d F at an arbitrary point  2 L1 ./ is given in the book [18], Sec. 3.4.2). Let  2 L1 ./ be an arbitrary tangent vector in the space T0 B./. Then for any sufficiently small t the function t 2 B./ and so determines the corresponding normalized quasiconformal homeomorphism w t , for which the following decomposition is true (2.6) w t .z/ D z C t w1 .z/ C o.t / for t ! 0 where o.t /  t ".z; t / and ".z; t / ! 0 uniformly on compact subsets of C. (This fact follows from the theorem on the uniform dependence of the solution w  of Beltrami equation on a parameter and is proved in [1], Ch. V, Sec. C.) The coefficient P (2.7) w1 .z/  wŒ.z/ coincides with the first variation of quasiconformal homeomorphism w  with respect to . We plug w t into the Beltrami equation and differentiate the obtained relation with respect to t for t D 0. Here one should use the fact that the operator @=@t commutes with the operators @N and @ (this fact is also proved in the book [1], Ch. V, Sec. C). We obtain the following relation N t D t@w t ; @w which implies that

 ˇ @ ˇˇ  N t  @w D  @w t ˇ tD0 : ˇ @t tD0 It follows form the decomposition (2.6) that ˇ @w t ˇ tD0 D 1;

(2.8)

@ ˇˇ ˇ w t D w1 .z/; @t tD0

N so we get from the relation (2.8) the @-equation on the function w1 .z/: N 1 D ; @w

(2.9)

which is satisfied for almost all z 2 C. If the support of  is compact, a solution of this equation is given by the Cauchy–Green integral Z 1 ./  d d ;  D  C i ;  C z plus an arbitrary entire function. In the book [1], Ch. V, Sec. C, it is shown that this entire function may be only a linear one of the form A C Bz. The constants A and B are easy to find from the normalization conditions w t .0; t /  0;

w t .1; t /  1;

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2 Universal Teichmüller space

whence w1 .0/ D w1 .1/ D 0. From the latter relations we deduce that Z Z   1 1 ./ 1 1 ./ d d ; B D  d d ; AD  C   C 1  Z Z 1z ./ ./ z d d C d d :     C C 1 Hence, the desired solution of equation (2.9) is given by the formula Z   1 z1 z 1 w1 .z/ D  ./ d d C   C z  1 Z z.z  1/ ./d d D :  C .  1/.  z/ so that

A C Bz D

(2.10)

This formula was obtained under the assumption that  has a compact support in C. One can get rid of this restriction, using the argument from [1], Ch. V, Sec. C. We use the formula (2.10) to prove the following theorem. Theorem 16. The differential of the map F W B./ ! Q. / at the point   0 is a bounded linear operator d0 F W L1 ./ ! Q. /, given by the formula Z 6 ./ d d ; z 2  ;  D  C i 2 : d0 F Œ.z/ D   .  z/4 The operator norm of d0 F is bounded by an absolute constant (which is at most 96). Idea of the proof (for the complete proof see Sec. 3.4.5 in the book [18]). Fix a point z0 2  . It can be shown that the function w t .z/ is holomorphic both with respect to z and to t (hence, with respect to .z; t /) for sufficiently small jt j and jz  z0 j (cf. [18], Sec. 3.4.3). So the function '.t; z/ WD SŒw t .z/; which is the image of the function t under the Bers embedding, is holomorphic in the domain fjtj < g fjz  z0 j < ıg  C2 for sufficiently small and ı. Let us compute the derivative of this function in t for t D 0. To make the formulas shorter, we shall denote the derivative in t by “dot”, and derivative in z by “prime”. Then 

'P D

w 000 3 w 002  w0 2 w 02



D

w 03 wP 000  wP 0 w 02 w 000  3wP 00 w 02 w 00 C 6wP 0 w 0 w 00 : w 04

2.2 Lecture V. Properties of the universal Teichmüller space

31

For t D 0 we have w.z/  z which implies that w 0  1, w 00 D w 000  0. So @' ˇˇ w 03 wP 000 D wP 000 : D ˇ @t tD0 w 04 We use now the formula (2.10) for the function w1 .z/ D wŒ.z/: P Z z.z  1/ ./d d w.z/ P D ; z 2 C; 

.  1/.  z/ (recall that we extend  by zero outside  while constructing the normalized solution w t ). The integral in the last formula is absolutely convergent so it defines a holomorphic function in z. Moreover, we can differentiate in z under the integral sign for z 2  since the resulting integral is uniformly bounded on compact subsets of  (cf. the estimate below). After the differentiation of the last formula in z, we obtain Z ./ 6 000 wP .z/ D  d d ; z 2  :  .  z/4 The estimate of the norm of the operator d0 F follows from the estimate of the integral Z  jd d j  for z 2  :  4 dist.z; S 1 /2

j  zj

We describe now the kernel of the differential d0 F . For that we introduce the subspace A2 ./  Q./, consisting of L1 -integrable holomorphic quadratic differentials in the disk : Z n o 2 A2 ./ D D .z/dz 2 Q./ W j .z/jjdxdyj < 1 :

There is a natural pairing between this space and the space B./ of Beltrami differentials Z h; i D

 :

This is a usual pairing between .1; 1/- and .2; 0/-differentials, given by the integration of an integrable .1; 1/-form on the right-hand side. In terms of this pairing the kernel d0 F is described in the following way. Lemma 6 (Teichmüller lemma). The kernel of the differential d0 F coincides with the subspace N WD A2 ./? D f 2 L1 ./ W h; i D 0 for all

2 A2 ./g:

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2 Universal Teichmüller space

Proof. We want to describe the set of all  2 L1 ./ such that Z ./ '.z/ WD d d D 0 for all z 2  : 4

.  z/ Consider the decomposition of the function 1=.  z/4 under the integral sign into the Taylor series in powers of 1=z in a neighborhood of the point z D 1 and integrate it with respect to . We get: c3 c4 '.z/ D 4 C 5 C    z z where the coefficients cn are computed by the formula Z cn D n.n  1/.n  2/  n3 ./d d for n D 3; 4; : : : :

From the condition that '.z/  0 for z 2  it follows that Z  k ./d d D 0 for k D 0; 1; 2; : : : ;

i.e. the function ./ is orthogonal to all powers  k , k D 0; 1; 2; : : : , hence, to all holomorphic polynomials. Since these polynomials are evidently dense in A2 ./, we have h; i D 0 for all 2 A2 ./.  2.2.5 Kähler structure of T . The results obtained in the previous section suggest an idea of how one can try to introduce a Kähler metric on the space T . Let us use for that the Ahlfors map (cf. [1], Ch. IV, Sec. D) ˆ W L1 ./ ! Q./; assigning to a function  2 L1 ./ the integral Z ˆŒ.z/  '.z/ D

./ d d : N 4 .1  z /

(2.11)

The image of the function  under this map is a holomorphic quadratic differential ' D '.z/dz 2 in the disk . The kernel of the map ˆ coincides with N D A2 ./? . We would like to define an Hermitian metric on the space T with the help of the formula (2.11). Let us try to define it first at zero, and then translate to other points of T with the help of the action of the group QS.S 1 / on T by left translations. So we define an Hermitian metric on the tangent space T0 T by setting it equal on tangent vectors Œ; Œ  2 T0 T D L1 ./=N to the double integral Z Z .; /  h; ˆŒ i D



.z/ ./ d d dxdy: .1  z/4

(2.12)

2.2 Lecture V. Properties of the universal Teichmüller space

33

However, the Hermitian metric on T0 T , introduced in this way, turns out to be correctly defined only on a dense subset in T0 T . The reason for that is that for a general

2 L1 ./ it may happen that its image ˆŒ  in Q./ is non integrable, i.e. it does not belong to A2 ./, and in this case the integral in the formula (2.12) will diverge. In fact, the formula (2.12) is correctly defined only for sufficiently smooth tangent vectors Œ, Œ  from T0 T . We formulate this assertion more precisely. Let Œ 2 L1 ./ be a tangent vector from T0 T . Consider the map   dˇ W T0 .B./= / ! TŒid QS.S 1 /=Möb.S 1 / ; (2.13) tangent to the isomorphism ˇ W B./=  ! QS.S 1 /=Möb.S 1 /: The tangent vector Œ under this map is sent to a vector field on S 1 of the form v. /

d d D wŒ.z/ P ; d dz

z D e i ;

where wŒ P is the first variation of the quasiconformal homeomorphism w  with respect to  from the formula (2.7). We shall call the vector fields on S 1 , which are the images of the elements Œ 2 T0 T under the map dˇ, quasisymmetric. We shall show below in Chapter 3 that the integral in formula (2.12) converges if the vectors , correspond to vector fields wŒ, P wŒ  P on S 1 from the smoothness class C 3=2C with any > 0. We return to the above correspondence (2.13) and give an intrinsic description of quasisymmetric vector fields on S 1 , being the images of vectors Œ 2 T0 .B./= / under the map (2.13). Since in this description the essential role is played by the Beurling–Ahlfors condition, it is convenient to start from the case of the upper halfplane H. The Zigmund space ƒ.R/ consists of continuous functions f W R ! R, satisfying the condition f .0/ D f .1/ D 0;

f .x/ !0 x2 C 1

for x ! 1;

for which there exists a constant C > 0 such that jf .x C t / C f .x  t /  2f .x/j  C jt j

for all x 2 R; t > 0:

The space ƒ.R/ is a (non separable) Banach space with the norm ˇ ˇ ˇ f .x C t / C f .x  t /  2f .x/ ˇ ˇ: kf kƒ WD sup ˇˇ ˇ t x;t In the paper [12] it is shown that quasisymmetric vector fields on R correspond to the functions from the Zigmund space ƒ.R/.

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2 Universal Teichmüller space

Having this description, it is easy to define an analogue of the space ƒ.R/ for the circle S 1 , using the Cayley transform. Namely, quasisymmetric vector fields on the d , where the function v W S 1 ! R is continuous, vanishes circle have the form v.e i / d at the points ˙1; i , and its image under the fractional-linear isomorphism  ! H is a function, satisfying the condition 

x2 C 1 xi v xCi 2



2 ƒ.R/:

The normalization condition v.˙1/ D v.i / D 0 is achieved by adding to the (non d normalized) vector field u.e i / d an appropriate vector field of the form .ae i C ae N i C b/

d ; d

where a 2 C, b 2 R, from the Lie algebra sl.2; R/ of the Lie group Möb.S 1 /. Brief content of Lecture V Metric and topological properties of universal Teichmüller space: Teichmüller distance is the distance between two normalized quasisymmetric homeomorphisms S 1 ! S 1 : dist.f; g/ D

1 log KŒg B f 1  2

where KŒ   is the maximal dilatation of a quasisymmetric homeomorphism. The space T , provided with the Teichmüller metric, is a contractible complete metric space. The Schwarzian derivative 1 SŒf  D .log f 0 /0  .log f 0 /2 2 has the following properties:  azCb D 0; 1) S czCd 2) S Œf  D S Œ1=f ; 3) S Œf B g D .SŒf  B g/ g 02 C SŒg; in the case when g is fractional-linear we have S Œf B g D .SŒf  B g/ g 02 , in the case when f is fractional-linear we have S Œf B g D SŒg; 4) for a given holomorphic function ' in a simply connected domain D there exists a function f , meromorphic in D, such that SŒf  D ':

2.2 Lecture V. Properties of the universal Teichmüller space

35

Bers embedding: This is an embedding of the universal Teichmüller space T into the space Q. / of holomorphic quadratic differentials in the complement  to the x given by the map closure of , Œ 7! SŒw  j  : This embedding induces on T a complex structure, which is compatible with the complex structure on this space, induced by the projection B./ ! T D B./=  : The composition of this projection with Bers embedding is a holomorphic map F W B./ ! Q. / with the differential at zero, equal to Z ./ 6 d0 F Œ.z/ D  d d ; z 2  ;  D  C i 2 :  .  z/4 The kernel of d0 F coincides with the subspace N WD A2 ./? D f 2 L1 ./ W h; i D 0 for all

2 A2 ./g

where A2 ./ is the space of integrable holomorphic quadratic differentials in the disk . A Kähler quasimetric on T is given on vectors Œ; Œ  2 T0 T D L1 ./=N by the formula .; / D h; ˆŒ i where ˆ W L1 ./ ! Q./ is the Ahlfors map, given by the formula Z ˆŒ.z/ D

./ d d : N 4 .1  z /

This quasimetric is correctly defined only on “smooth” tangent vectors Œ; Œ  2 T0 T , which correspond to the C 3=2C -smooth quasisymmetric vector fields on the circle S 1 . The space ƒ.R/ of quasisymmetric vector fields on the real line R coincides with the Zigmund space, consisting of continuous functions f W R ! R such that f .0/ D f .1/ D 0; and

f .x/ !0 x2 C 1

for x ! 1;

jf .x C t / C f .x  t /  2f .x/j < 1: t x2R; t>0 sup

3 Subspaces of universal Teichmüller space

In this chapter we study in detail the two kinds of submanifolds of universal Teichmüller space, namely, the classical Teichmüller spaces T .G/ and the space  of normalized diffeomorphisms of the circle. The classical Teichmüller spaces are considered in Lecture 3.2. It is preceded by Lecture 3.1, in which we present necessary facts from the theory of Riemann surfaces. In this lecture we follow the approach, based on the theory of Fuchsian groups, as the most convenient to define an embedding of classical Teichmüller spaces T .G/ into the universal Teichmüller space T . The spaces T .G/ are embedded into T as complex submanifolds. Moreover, the Kähler quasimetric on T , defined in Section 2.2.5, under this embedding is reduced to the Kähler Weil–Petersson metric on the spaces T .G/. The space of normalized diffeomorphisms of the circle  is also embedded into T as a complex submanifold. Moreover, in contrast with the classical Teichmüller spaces T .G/, the image of this embedding is contained in the regular part of the space T . The Kähler quasimetric on T , being restricted to , yields a genuine Kähler metric.

3.1 Lecture VI. Riemann surfaces 3.1.1 Kleinian groups. There exist different approaches to the definition of Riemann surfaces. The most convenient for our goals is the approach, based on the theory of Kleinian groups, in which a Riemann surface arises as the quotient of one of the standard x complex plane C or unit disk ) modulo complex domains (i.e. the Riemann sphere C, an appropriate discrete group of fractional-linear transformations. We start from the classification of such transformations. Suppose that a fractionallinear transform is given by   az C b a b w.z/ D where 2 SL.2; C/: c d cz C d Associate with this transform the quantity tr2 w D .a C d /2 2 C: This quantity is invariant under conjugation in the group Möb.C/, according to the following problem.

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3 Subspaces of universal Teichmüller space

Problem 6. The transformations u; v 2 Möb.C/, which are not equal to the identity transformation I , are conjugate to each other, i.e. u D gvg 1 for some g 2 Möb.C/ if and only if tr2 u D tr2 v. Classification of fractional-linear transformations. A fractional-linear transformation w is called parabolic if tr2 w D 4: x As This condition is equivalent to the condition that w has a unique fixed point on C. an example of such a transformation one may take the map: w.z/ D z C 1. A fractional-linear transformation w is called elliptic if tr2 w is real and 0  tr2 w < 4: This condition is equivalent to the condition that w is conjugate (in the group Möb.C/) to the rotation z 7! z with j j D 1: A fractional-linear transformation w is called loxodromic if it does not belong to the first two classes. In this case w is conjugate to the transformation z 7! z

with j j ¤ 1:

The most important subclass of loxodromic transformations is formed by the hyperbolic transformations which are conjugate to the transformations from the group Möb.R/ of fractional-linear transformations given by the formula   az C b a b w.z/ D where 2 SL.2; R/: c d cz C d Definition 8. Let G be a discrete subgroup in Möb.C/. The set x W G acts properly discontinuously at zg  D .G/ D fz 2 C is called the discontinuity domain of the group G. The group G acts properly discontinuously at a point z if the stabilizer Gz D fg 2 G W gz D zg of this point consists of finite number of elements and there exists a neighborhood U of this point such that g.U / D U for all g 2 Gz

and

g.U / \ U D ; for all g 2 G n Gz :

x It consists of The discontinuity domain .G/ is an open invariant subset in C. not more than a countable number of connected components and, in particular, may be empty.

3.1 Lecture VI. Riemann surfaces

39

Definition 9. A discrete subgroup G  Möb.C/ is called Kleinian if .G/ ¤ ;. The stabilizer Gz of the Kleinian group at a point z 2 .G/ may consist only of a unit and elliptic elements of finite orders. The complement of .G/ is called the limit set of the group G: x n .G/: ƒ.G/ WD C It coincides with the set of limit points of the group G. The limit set ƒ.G/ of a Kleinian group may be finite (more concretely, empty or containing one or two points) or infinite (and in this case not countable). In the case of a finite ƒ.G/, i.e. when ƒ.G/ is empty or consists of one or two points, the group G is called elementary. In the case of uncountable ƒ.G/ the group x i.e. it is closed, G is called non-elementary, in this case ƒ.G/ is a perfect set in C, everywhere dense in itself and has an empty interior (in particular, no isolated points). Moreover, in this case ƒ.G/ coincides with the set of limit points of any orbit. Definition 10. A Kleinian group G is called Fuchsian if its limit set lies on a circle in x (i.e. on a line or circle in C), dividing C x in two disks, each of them being preserved C by the group G. Using conjugation, we can always reduce to the case when this circle coincides with x  C. x We shall assume that this condition is satisfied. In this case the group G acts R properly discontinuously both on the upper halfplane H D HC and lower halfplane H . Note that any discrete subgroup of Möb.R/ acts totally discontinuously on H, hence is a Fuchsian group. (However not every discrete subgroup in Möb.C/ is even a Kleinian one.) Definition 11. A Fuchsian group G is called a Fuchsian group of the first kind if x In this case ƒ.G/ D R. .G/ D HC [ H : A Fuchsian group G is called a Fuchsian group of the second kind if x g: .G/ D HC [ H [ fa proper open subset in R In this case the discontinuity domain .G/ is connected. Definition 12. The fundamental domain of a Kleinian group G is an open set D  .G/, having the following properties: 1) different points of D belong to different orbits; x 2) the orbit of any point z 2 .G/ intersects D; x n D is equal to 3) (two-dimensional) Lebesgue measure of the boundary @D D D zero.

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3 Subspaces of universal Teichmüller space

This definition is immediately carried over to the case when G acts not on the whole domain .G/, but on some of its invariant subsets. For example, in the case of a Fuchsian group G  Möb.R/ it is natural to look for the fundamental domain for the action of G on the upper halfplane H. In this case we denote by U.G/ the maximal domain in H, on which the group G acts freely, i.e. without fixed points. Equivalently, U.G/ consists of the points from H which are not fixed by elliptic elements from G. In such a situation we can take as a fundamental domain for the action of G on H the hyperbolic polygon with center at a point z0 2 U.G/: D D fz 2 H W d.z; z0 /  d.z; g.z0 // for all g 2 Gg where d is the hyperbolic metric on the upper halfplane H, defined by the formula ds 2 D

dx 2 C dy 2 4y 2

for x C iy 2 H:

This polygon is also called the Dirichlet fundamental domain. In the case of a finitely generated group G the boundary of this polygon consists of a finite number of segments of hyperbolic geodesics. The edges of D are divided into pairs in @D – each of them is conjugate to another edge (in particular, it can be conjugate to itself) obtained from the first one by the action of some element g 2 G. In order to obtain from D a model of the Riemann surface H=G, it is sufficient to glue @D along the conjugate edges. If G is a Fuchsian group of the first kind then the fundamental domain D may touch x only in a finite number of parabolic points (“cusps”). R Examples of Fuchsian and Kleinian groups 1. If the limit set ƒ.G/ of a Fuchsian group G is empty then the group G is necessarily finite and coincides with a cyclic group, consisting of elliptic elements. 2. If the limit set ƒ.G/ contains only one point then G is the infinite cyclic group with a parabolic generator. 3. If the limit set ƒ.G/ consists of two points then G is the infinite cyclic group with a hyperbolic generator. The limit points in these cases are fixed. 4. The torus group is generated by two complex numbers 1 ; 2 , one of which may be taken real and another lying in the upper halfplane H. The group .1 ; 2 / D fz 7! z C m1 C n2 W .m; n/ 2 Z2 g is an elementary Kleinian group with ./ D C, ƒ./ D f1g. 5. The elliptic modular group G1 D PSL.2; Z/ WD SL.2; Z/=f˙I g

41

3.1 Lecture VI. Riemann surfaces

is a classical example of a Fuchsian group. Along with this group we can consider the modular groups of level k: ²



a b Gk D g D c d



 2 G1 W

a b c d





 ³ 1 0  mod k 0 1

All these groups Gk are finitely generated Fuchsian groups of the first kind. 3.1.2 Riemann surfaces. Recall the standard construction of compact Riemann surfaces of genus g. The fundamental group 1 .X / D 1 .X; x0 / of such a surface X is generated by 2g loops A1 ; B1 ; : : : ; Ag ; Bg with the unique relation g Y

ŒAi ; Di  D 1

iD1 1 where ŒAi ; Di  WD Ai Bi A1 i Bi .

Definition 13. A Riemann surface has a (finite) conformal type .g; n/ if it is biholomorphic to a surface of the form Xy n fx1 ; : : : ; xn g where Xy is a compact Riemann surface of genus g, and x1 ; : : : ; xn are pairwise disjoint y points on X. Such a surface is also called the Riemann surface with n punctures. Its fundamental group is generated by 2g loops A1 ; B1 ; : : : ; Ag ; Bg and n “small” loops C1 ; : : : ; Cn around the punctures x1 ; : : : ; xn . The only relation between the generators has the form g Y ŒAi ; Di   C1 : : : Cn D 1: iD1

Uniformization. The Uniformization Theorem asserts that the universal cover Xz of x or the complex plane C, or the a Riemann surface X is either the Riemann sphere C, upper halfplane H (or, equivalently, the unit disk ). In other words, the surface X is biholomorphically equivalent to the surface Xz =G where G Š 1 .Xz / is a discrete group of biholomorphic automorphisms of the surface z acting freely and properly discontinuously on Xz . X, A Riemann surface X1 D Xz =G1 is biholomorphically equivalent to a Riemann surface X2 D Xz =G2 if and only if the groups G1 and G2 are conjugate inside the automorphism group Aut.Xz / (prove it!).

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3 Subspaces of universal Teichmüller space

Correspondence: Riemann surfaces ! Kleinian groups. Let G be a discrete Kleinian subgroup in the group Möb.C/. Then .G/=G  =G is the union of (at most) a countable number of Riemann surfaces so that the projection  W  ! =G is an open holomorphic map. Indeed, if G acts freely on  then  is a covering, inducing the structure of a complex manifold on =G. If the stabilizer Gz0 of some point z0 2  is non trivial then Gz0 is a cyclic group of a finite order k0 . Denote its elliptic generator by g0 . Then in terms of an appropriate local coordinate w in a neighborhood of z0 the action of g0 will be given by the formula g0  w D e 2 i=k0 w: In this case the projection  is a k-fold covering in a truncated neighborhood of z0 and the function w k may be taken for the local holomorphic coordinate in a neighborhood of the point .z0 / 2 =G. The point .z0 / is a branching point of kth order, and the projection  over a neighborhood of the point .z0 / is a holomorphic branched covering of degree k. The obtained Riemann surfaces (connected components of =G) may have any conformal type .g; n/, apart from .0; 0/, .0; 1/, .0; 2/ and .1; 0/. The latter types x and complex plane correspond to the Riemann surfaces, having the Riemann sphere C C as their universal covers. More precisely, the type .0; 0/ corresponds to the Riemann x covered by itself. The types .0; 1/, .0; 2/ and .1; 0/ correspond respectively sphere C, to the complex plane C, truncated complex plane C n f0g and the torus. All these Riemann surfaces are covered by the complex plane C. If G is a Fuchsian group of the first kind then X D =G D XC [ X where X˙ D H˙ =G. The Riemann surfaces X˙ are dual to each other and the complex conjugation determines an antiholomorphic homeomorphism of XC onto X . If the group G is a Fuchsian group of the second type, i.e. the discontinuity domain .G/ is connected, then the Riemann surfaces X˙ are biholomorphically embedded into the Riemann surface X d D =G; called the Schottky double of the surface XC so that the images of X˙ in X d are glued together along their common boundary x @XC D @X D . \ R/=G: The antiholomorphic involution on X d , induced by the complex conjugation z 7! zN on , interchanges XC and X , leaving the points of their common boundary fixed.

3.1 Lecture VI. Riemann surfaces

43

Riemann surfaces, covered by the upper halfplane. Let G be now a finitely generated Fuchsian group of the first kind. Then after the factorization of the upper halfplane H modulo G we shall obtain a Riemann surface X of a finite conformal type. If there are no elements of finite orders in G then H is the universal cover of the Riemann surface X with the fundamental group 1 .X / D G. If G has elements of finite orders then we consider first the Riemann surface U.G/=G where U.G/ is the maximal domain in H, on which the group G acts freely. Then U.G/ ! U.G/=G is a (non-ramified) holomorphic covering, and U.G/=G is a Riemann surface of conformal type .g; nCm/ with nCm punctures where these punctures are of the following form. The first n of them correspond to elliptic points z1 ; : : : ; zn of orders 1 ; : : : ; n respectively so that the stabilizer Gzj of the point zj , j D 1; : : : ; n, coincides with the cyclic group of order j . The Riemann surface X D H=G is obtained from U.G/=G by filling up these punctures with the branching points of orders 1 ; : : : ; n . The rex The maining m punctures correspond to the parabolic points (“cusps”), lying on R. stabilizer of each of these points is the infinite cyclic group with a parabolic generator. The Riemann surface X D H=G in this case will have conformal type .g; m/. Brief content of Lecture VI azCb Classification of Möbius transformations: A transformation w D czCd is called 2 parabolic if tr w D 4, for example, z 7! z C 1; such transformations have a unique fixed point. A transformation w is called elliptic if 0  tr2 w < 4, for example, z 7! z with j j D 1. A transformation w is called loxodromic if it does not belong to the two considered types, for example, z 7! z with j j ¤ 1. Transformations from the group Möb.R/ of real fractional-linear transformations are called hyperbolic. x at which the discrete group G of fractional-linear The set .G/ of points z 2 C, transformations acts properly discontinuously, is called the discontinuity domain. Its x n .G/ is called the limit set of the group G. The group G is complement ƒ.G/ D C called Kleinian if .G/ ¤ ;. The limit set of such group is either empty, or finite and consists of one or two points, or non-countable. A discrete group G of fractional-linear transformations is called Fuchsian if ƒ.G/  x The Fuchsian groups of the first kind are singled out by the condition: ƒ.G/ D R, x R. x The group and Fuchsian groups of the second kind by the condition ƒ.G/ ¤ R. PSL.2; Z/ of integer fractional-linear transformations may serve as an example of Fuchsian groups of the first kind. If G is a Kleinian group then .G/=G is the union of Riemann surfaces, and the projection .G/ ! .G/=G is a holomorphic covering. For a Fuchsian group of the first kind .G/ D XC [ X where X˙ D H˙ =G are the quotients of the upper (resp. lower) halfplane modulo the action of the group G. For a Fuchsian group of the second

44

3 Subspaces of universal Teichmüller space

kind X˙ are the subsets of the Riemann surface X d D .G/=G with the common boundary @XC D @X . Riemann surfaces of type H=G with Fuchsian group G of the first kind: If the group G has no torsion then X D H=G is a Riemann surface and the projection H ! H=G D X is a holomorphic covering. In the case when G has elements of finite orders, we consider the maximal subdomain U.G/ in H, on which the group G acts freely; then the projection U.G/ ! U.G/=G determines a holomorphic covering of the Riemann surface U.G/=G of conformal type .g; n C m/ (i.e. Riemann surface of genus g with n C m punctures) such that the first n of these punctures correspond to elliptic points on H, and another m x Filling up the first n punctures by ellippunctures correspond to parabolic points on R. tic branching points of appropriate orders, we obtain the Riemann surface X D H=G of conformal type .g; m/.

3.2 Lecture VII. Classical Teichmüller spaces 3.2.1 G -invariant quasiconformal maps Definition 14. A quasiconformal homeomorphism w is called G-invariant with respect to a Kleinian group G (or compatible with the action of G) if wGw 1  Möb.C/: In other words, the group wGw 1 is again Kleinian and its discontinuity domain coincides with w ..G//. This definition admits the following reformulation in terms of Beltrami differentials. x with Proposition 1. A quasiconformal homeomorphism w of the Riemann sphere C complex dilatation  is G-invariant if and only if the following condition .gz/

g 0 .z/ D .z/ g 0 .z/

(3.1)

x and all g 2 G. is satisfied for all almost z 2 C Proof. It is sufficient to prove this assertion for the normalized homeomorphism w  , since it differs from w only by a fractional-linear transformation. For w  its Ginvariance means that for any g 2 G the following relation w  B g B .w  /1 DW g  2 Möb.C/ H) w  B g D g  B w  holds. Differentiating the latter equality in z, N we get     N  B g/ D @w N  B g gN 0 D .g  /0 B w  @w N ; @.w

3.2 Lecture VII. Classical Teichmüller spaces

45

while its differentiation with respect to z yields  @.w  B g/ D .@w  B g/ g 0 D .g  /0 B w  @w  : Dividing the first relation by the second one, we obtain . B g/

gS0 D g0

x Hence, the condition (3.1) is necessary for the G-invariance almost everywhere on C.  of w and w. By reversing all implications we see that this condition is also sufficient for the G-invariance of w  and w.  The given definition and the proved proposition extend immediately to quasiconformal homeomorphisms, defined in the unit disk . If the transformations from the group G preserve  then the quasiconformal homeomorphisms w  and w , associated with a G-invariant quasiconformal homeomorphism w of the disk , will be G-invariant in x since the extensions of Beltrami differential  by zero and by reflection with respect C, to S 1 do not violate the condition (3.1). Hence we can associate with any Beltrami differential  in the disk , invariant under a Fuchsian group G and preserving , the Kleinian groups 1 G WD w Gw

and

G  WD w  G.w  /1 :

Moreover, the group G is again a Fuchsian group, since it preserves the disk . 3.2.2 Classical Teichmüller spaces. Let G  Möb.S 1 / be a Fuchsian group of the first kind. The definition of the G-invariance of quasiconformal homeomorphisms extends immediately to quasisymmetric homeomorphisms of S 1 . Definition 15. A quasisymmetric homeomorphism f 2 QS.S 1 / is called G-invariant with respect to a Fuchsian group G (or compatible with the action of G) if f Gf 1  Möb.S 1 /: Denote the subgroup of G-invariant quasisymmetric homeomorphisms in QS.S 1 / by QS.S 1 /G and define the classical Teichmüller space T .G/ as T .G/ D QS.S 1 /G =Möb.S 1 /: This space can be defined in terms of Beltrami differentials as T .G/ D B./G =  where B./G is the subspace of G-invariant Beltrami differentials in B./, consisting of differentials  2 B./, satisfying the relation (3.1), while the equivalence relation

46

3 Subspaces of universal Teichmüller space

remains the same, as in the case of universal Teichmüller space, i.e.   , w D w on  . The Teichmüller space T .G/ is a complex Banach manifold with the complex structure being induced by the Bers embedding or the natural projection B./G ! T .G/ D B./G = . Thus, all classical Teichmüller spaces T .G/ are embedded into the universal Teichmüller space T , associated with the Fuchsian group f1g, as complex submanifolds. Let a Riemann surface X be uniformized by a Fuchsian group G, i.e. X D =G: We can associate with every class Œ 2 T .G/ a new Riemann surface X D =G 1 where G D w Gw . The same surface may be written in the form (check the validity of this assertion!) X D  =G 

where  WD w  ./, G  D w  G.w  /1 (note that G  acts on  properly discon   tinuously). At the same time the Riemann surface   =G , where  WD w . /, is  biholomorphically equivalent to  =G, since w is conformal in  . In other words, the space T .G/ parameterizes, by assigning Œ 7! G , different complex structures on a given topological surface X D =G which can be obtained from the original one by quasiconformal deformations. (For a more detailed discussion cf. [1], Ch. VI, Sec. A.) All properties of the universal Teichmüller space, presented in Chapter 2, carry on to the classical Teichmüller spaces, it is necessary only to add every time the G-invariance condition. For example, the Bers embedding in the case of the unit disk  is given by the map F W B./G ! Q. /G ; associating with a Beltrami differential  2 B./G the holomorphic quadratic differential S Œw  j   in  . The space Q. /G by definition consists of G-invariant holomorphic quadratic differentials in  , having the finite norm k k2 WD sup .1  jzj2 /2 j .z/j < 1: z2 

The formula for the differential d0 F has the same form: Z ./ 6 d0 F Œ.z/ D  d d ;  C .  z/4

z 2  ;

for  2 L1 .C /G . The kernel of d0 F coincides with the subspace N G  .A2 .C /G /? D f 2 L1 .C /G W h; i D 0 for all

2 A2 .C /g:

3.2 Lecture VII. Classical Teichmüller spaces

47

So, the tangent space to T .G/ at the origin coincides with L1 ./G =N G . As in Chapter 2, there is the Ahlfors map L1 ./G =N G ! Q./G given by the formula Z ./ d d : L1 ./G 3  7! ˆŒ.z/ D N 4

.1  z / Let us try, as in Chapter 2, to introduce with the help of this map a Kähler metric on the space T .G/ by defining it on vectors Œ, Œ , belonging to T0 T .G/ D L1 ./G =N G , by the formula Z

Z

.; /G D h; ˆŒ iG WD

=G



.z/ ./ d d dxdy: N 4 .1  z /

(3.2)

In the case of classical Teichmüller spaces T .G/, i.e. under the condition that our Riemann surface =G has a finite conformal type, the space Q./G coincides with the space of integrable holomorphic quadratic differentials A2 ./G (cf. [18]), which implies that the formula (3.2) is correctly defined for all ; 2 L1 ./G and determines a Hermitian metric on the space T0 T .G/, called the Weil–Petersson metric. What can be said on the image of a classical Teichmüller space T .G/ in the universal Teichmüller space T ? There is an interesting result by Bowen [7], showing that this image does not belong to the regular part of T . More precisely, we shall call a point of T regular if it is associated with a smooth normalized quasisymmetric homeomorphism from QS.S 1 / or, equivalently, with a quasidisk with a smooth boundary. For classical Teichmüller spaces T .G/ where G is the Fuchsian group, uniformizing the compact Riemann surface X D =G, Bowen has proved the following theorem. He has shown that every point from T .G/ n f0g is associated with a quasidisk with fractal boundary, having the Hausdorff dimension dH in the interval 1 < dH < 2 (and may take any value from this interval). In terms of quasisymmetric homeomorphisms it may be proved that if f is a G-invariant quasisymmetric homeomorphism belonging to the class C 1 at one point, then f 2 Möb.S 1 /. In the next lecture we shall study another subspace of T lying entirely in its regular part. Brief content of Lecture VII A quasiconformal homeomorphism w is called G-invariant with respect to a Kleinian group G if wGw 1  Möb.C/: Analogously, a quasisymmetric homeomorphism f is called G-invariant with respect to a Fuchsian group of the first kind G if f Gf 1  Möb.S 1 /:

48

3 Subspaces of universal Teichmüller space

A Beltrami differential  is called G-invariant with respect to the group G if .gz/

g 0 .z/ D .z/: g 0 .z/

The Teichmüller space associated with a Fuchsian group of the first kind G, is defined as T .G/ D QS.S 1 /G =Möb.S 1 / D B./G =  where QS.S 1 /G denotes the subgroup of G-invariant quasisymmetric homeomorphisms in QS.S 1 /, and B./G is the subspace of G-invariant Beltrami differentials in B./. If X D =G is a Riemann surface, uniformized by the unit disk , then we can associate with any Beltrami differential  2 B./ a new Riemann surface X D =G D  =G  1 where G D w Gw , G  D w  G.w  /1 , and the quasidisk  is defined as    WD w ./. This surface is biholomorphic to X ()   0. So, the space T .G/ parameterizes, with the help of the correspondence Œ 7! G , different complex structures on the topological Riemann surface X D =G, which are obtained from the original one by quasiconformal deformations.

The Weil–Petersson Kähler metric on T .G/ is defined on vectors Œ; Œ  2 T0 T .G/ by the formula Z Z .z/ ./ d d dxdy: .; /G D N 4

=G .1  z / The space T .G/ n f0g is contained in the non-regular part of the universal Teichmüller space T contrary to the subspace studied in the next lecture.

3.3 Lecture VIII. The space of normalized diffeomorphisms  D DiffC .S 1 /=Möb.S 1 / 3.3.1 Complex structure. The space  D DiffC .S 1 /=Möb.S 1 /  T D QS.S 1 /=Möb.S 1 /

(3.3)

of normalized diffeomorphisms, introduced in Section 2.1.1, is contained in the regular part of the space T . The embedding (3.3) provides  with the induced complex structure. However, this complex structure on  may be also introduced in a more direct way. We note, first of all, that it is sufficient to define this complex structure only at the origin Œid 2 , and then transmit it to other points of  with the help of the action of the

3.3 Lecture VIII. The space of diffeomorphisms

49

group DiffC .S 1 /. The complex structure constructed in this way will be automatically DiffC .S 1 /-invariant. The tangent space   TŒid  D TŒid DiffC .S 1 /=Möb.S 1 / may be identified with the quotient of the Lie algebra of the Lie group DiffC .S 1 / modulo its subalgebra sl.2; R/, coinciding with the Lie algebra of the Lie group Möb.S 1 /. The Lie algebra of the Lie group DiffC .S 1 / coincides with the Lie algebra Vect.S 1 / of smooth vector fields on S 1 , and it is convenient to represent its elements v D v. /@=@ by the Fourier decompositions of the form X vD vn en n2Z

with complex coefficients, satisfying the condition vN n D vn , where en are the basis vector fields en D e i n @=@ D iz nC1 @=@z; n 2 Z; z D e i : Then the elements v 2 TŒid  will be given by the series of the form X vD vn en : n¤0;˙1

The complex structure J on the space TŒid  is defined by the formula J v D i

1 X nD2

vn en C i

1 X

vn en :

(3.4)

nD1

Let us compare the introduced complex structure with the structure I on the space TŒ0 T constructed above, using the realization T D B./= , and show that these structures are equivalent. In terms of Beltrami differentials the action of the complex structure I corresponds to the multiplication of the Beltrami differential  by i . Recall (cf. 2.2.5) that the Beltrami differential  is associated with the quasisymmetric vector field on S 1 , having the form v D wŒ@=@z P where (cf. Section 2.2.4) wŒ P is the first variation of the quasiconformal homeomorphism w t with respect to , defined by the decomposition P C o.t / w t .z/ D z C t wŒ.z/

for t ! 0

which has the same sense, as in (2.6). In order to compute the complex structure J , we should for a given vector field v D v. /@=@ D izv.z/@=@z D wŒ@=@z P

50

3 Subspaces of universal Teichmüller space

find a vector field J v D v. Q /@=@ D iz v.z/@=@z Q D wŒi@=@z: P The transform, assigning to the function v.z/ the function v.z/, Q is called the Hilbert transform. We shall find its explicit form a little bit later. N The variation wŒ, P as in Section 2.2.4, satisfies the @-equation @N wŒ P D

almost everywhere in :

(3.5)

Hence the function F WD wŒi P  i wŒ P is holomorphic in the disk , and on the circle S 1 it coincides with F .z/ D iz v.z/ Q C zv.z/;

z D e i :

(3.6)

Note that the boundary value of the function F on S 1 is correctly defined, since N the Cauchy–Green integral of the function  2 L1 ./, giving a solution of the @1 equation (3.5) in the disk , is Hölder-continuous up to S . By writing the function F in the form F .z/ D F .0/ C zF1 .z/, we shall obtain from the formula (3.6) the relation Q C v.z/; zF N .0/ C F1 .z/ D i v.z/

z D e i :

Taking into account the real-valuedness of the functions v. / and v. Q /, we deduce the following relations:   v D Re.NzF .0/ C F1 .z// D Re zF .0/ C F1 .z/ D Re f .z/;   vQ D Im .Nz F .0/ C F1 .z// D Im zF N .0/  zF .0/ C f .z/ D Im f .z/ C bz C bN zN where b D iF .0/ and f .z/ D zF .0/ C F1 .z/ is a holomorphic function in the disk  such that its real part on the circle S 1 coincides with v: Re f . / D v. /. Such a function f is uniquely defined by the function v up to an imaginary constant and is given by the Schwarz formula: f .z/ D

1 2

Z

2 0

z C e i v. /d C i a z  e i

where a 2 R:

(3.7)

It follows from these relations and the Schwarz formula(3.7) that v.z/ Q D Im f .z/ C bz C bN zN C a

where a 2 R; b 2 C:

If the function v. / is given by the Fourier decomposition of the form X vn e i n v. / D n2Z

(3.8)

3.3 Lecture VIII. The space of diffeomorphisms

51

then the function f , holomorphic in the disk  and satisfying the condition Re f D v on S 1 , will be given by the power series f .z/ D v0 C 2

1 X

vn z n :

nD1

So the representation (3.8) implies that N i C a v. Q / D Im f .e i / C be i C be 1 1 X X N i /: D .ivn /e i n C .ivn /e i n C .˛ C ˇe i C ˇe nD2

(3.9)

nD2

Setting ˛ D ˇ D 0 in this relation, we can choose the constants a, b in such a way that the component of v, Q belonging to sl.2; R/, will vanish. Then for vQ D J v we shall obtain the formula (3.4). Hence, we have shown that the two definitions of the complex structure J on , given above, are equivalent which implies that the embedding of the manifold , provided with the complex structure J , into the universal Teichmüller space T , provided with the complex structure I , is holomorphic. 3.3.2 Kähler metric. The space  has a homogeneous symplectic form !. This form is uniquely determined, up to multiplication by a constant, by its values on the basis C vector fields en 2 TŒid  which are equal to (cf. [27], Sec. 10.3): !.em ; en / D ˛.m3  m/ım;n ;

m; n 2 Z n f0; ˙1g; ˛ 2 C n f0g:

From the form ! and complex structure J we can construct a Riemannian metric gR , compatible with these structures. The value of this metric on the tangent vectors u; v 2 TŒid  is given by the formula gR .u; v/ D a Re where u D

P n¤0;˙1

1 hX

i uN n vn .n3  n/ ;

a 2 R n f0g;

nD2

un en , v D

P

n¤0;˙1

g.u; v/ D a

vn en . The Kähler metric

1 X

uN n vn .n3  n/

(3.10)

nD2

coincides with the complexification of the constructed Riemannian metric gR . Note that the series on the right-hand side of (3.10) is absolutely convergent if the vector fields u; v belong to the smoothness class C 3=2C with any > 0. The Kähler metric (3.10) coincides with the restriction to  of the Kähler quasimetric (2.12) from Section 2.2.5 for the appropriate choice of the constant a. More precisely, we have the following

52

3 Subspaces of universal Teichmüller space

Proposition 2. Assume that the vectors ; 2 L1 ./ correspond to C 3=2C -smooth vector fields u, v on S 1 respectively. Then the value of the metric (3.10) on these fields will be equal to Z Z a .z/ ./ g.u; v/ D dx dy: N 4 6 2 .1  z / Idea of the proof (for the complete proof see [21]). As in the proof of the Teichmüller lemma from Section 2.2.4, one can show, using the integral representation for wŒ P (formula (2.10)) that the Fourier coefficients of the vector field u D u. /@=@ may be computed by the formula Z i un D zN n2 .z/dxdy for n  2 

and uN n D un for n  2. It implies that 1 X

1 uN n vn .n  n/ D  2  nD2

Z Z

3

.z/ ./



1 X

z n2 N n2 .n3  n/dxdy d d :

nD2

(3.11) (We have interchanged the integration and summation signs, using the uniform convergence of the considered series for C 3=2C -smooth vector fields.) It remains to take into account that 1 X

n2 .n3  n/ D 

nD2

1 6.1  /4

for j j < 1

N and substitute this equality into the formula (3.11) for D z .



The image of the embedding  ,! T lies, as it was already pointed out before, in the regular part of the space T . On the other hand, the images of the embeddings of the classical Teichmüller spaces T .G/ n f0g belong to the non-regular part of T . So, these submanifolds intersect only at the origin Œid 2 T . Brief content of Lecture VIII The space of normalized diffeomorphisms:  D DiffC .S 1 /=Möb.S 1 /  T D QSC .S 1 /=Möb.S 1 /: P The complex structure at the origin is given on a vector v D n¤0;˙1 vn en 2 TŒid  by the formula 1 1 X X vn en C i vn en : J v D i nD2

nD1

This structure coincides with the complex structure, induced by the embedding   T .

3.3 Lecture VIII. The space of diffeomorphisms

53

The symplectic structure at the origin, defined up to a multiplicative constant, is given on basis vectors en by the formula !.em ; en / D ˛.m3  m/ım;n : The compatible Kähler metric on vectors X X C uD un en ; v D vn en 2 TŒid  n¤0;˙1

n¤0;˙1

is given by the formula g.u; v/ D a

1 X

uN n vn .n3  n/:

nD2

This metric (for a D 6 2 ) coincides with the metric, induced by the quasimetric on T under the embedding   T , and is given on vector fields u, v, corresponding to Beltrami differentials ; , by the formula Z Z



.z/ ./ dx dy: N 4 .1  z /

The space  D DiffC .S 1 /=Möb.S 1 / belongs to the regular part of the universal Teichmüller space T and intersects with the classical Teichmüller spaces T .G/ only at the origin.

4 Grassmann realization of the universal Teichmüller space

In this chapter we construct a realization of the universal Teichmüller space T as a submanifold of the infinite-dimensional Grassmannian which plays a key role in the quantization of T . It is based on the Nag–Sullivan theorem, proved in Section 4.1.3. This theorem asserts that the group of quasisymmetric homeomorphisms of the circle QS.S 1 / acts on the Sobolev space of half-differentiable functions on the circle V WD H01=2 .S 1 ; R/ by symplectic transformations. We review the properties of the Sobolev space V in Section 4.1.1 and Section 4.1.2. The Nag–Sullivan theorem implies that the universal Teichmüller space is embedded into the space J.V / of complex structures on the Sobolev space V , compatible with the symplectic structure. The latter space in its turn may be identified with the infinitedimensional Siegel disk D, embedded into the infinite-dimensional Grassmannian of the Hilbert space V . The resulting embedding of the universal Teichmüller space T into this Grassmannian is a holomorphic map of complex Banach manifolds. The restriction of the above embedding to the space  of normalized diffeomorphisms of the circle realizes this space as a submanifold of the Hilbert–Schmidt Grassmannian GrHS .V C / of the Hilbert space V C .

4.1 Lecture IX. The action of quasisymmetric homeomorphisms on the Hilbert space 4.1.1 The Sobolev space of half-differentiable functions Definition 16. The Sobolev space of half-differentiable functions V D H01=2 .S 1 ; R/ is the Hilbert space consisting of functions f 2 L2 .S 1 ; R/ with zero average around the circle, having generalized derivative of order 1/2 in L2 .S 1 ; R/. In other words, it consists of functions f 2 L2 .S 1 ; R/ with Fourier series of the form X f .z/ D fn z n ; fNn D fn ; z D e i ; n¤0

56

4 Grassmann realization of the universal Teichmüller space

with finite Sobolev norm of order 1/2 kf k21=2 D

X

1 X

jnjjfn j2 D 2

njfn j2 < 1:

nD1

n¤0

By assigning to a function f 2 V the sequence ffn g, n D 1; 2; : : :, of its Fourier coefficients, we establish an isometric isomorphism between the space V and the Hilbert E space `1=2 2 , consisting of sequences f D .f1 ; f2 ; : : :/ of complex numbers fn with finite norm 1 X njfn j2 < 1: kfEk21=2 WD 2 `2

nD1

Consider on the space V a skew-symmetric 2-form ! W V V ! R which is defined in terms of Fourier coefficients of vectors ; 2 V by the formula !.; / D i

X

nn n D 2 Im

1 X

nn N n :

nD1

n¤0

This form is correctly defined by the Cauchy–Schwarz inequality j!.; /j  kk1=2  k k1=2 : The form ! coincides with the extension of the natural symplectic structure on the space of smooth loops S 1 D C01 .S 1 ; S 1 /; given by the 2-form 1 !0 .; / D 2

Z

2

.e i / 0

d .e i / d d

(4.1)

(cf. [27], Sec. 7.2). The space V also has a complex structure J 0 which is defined in terms of Fourier decompositions by the formula .z/ D

X

n z n 7! .J 0 /.z/ D i

1 X

n z n C i

n z n :

(4.2)

nD1

nD1

n¤0

1 X

This complex structure is compatible with the symplectic form ! in the sense that they define together a Riemannian metric on V by the formula g 0 .; / WD !.; J 0 /, or in terms of Fourier coefficients g 0 .; / D

X n¤0

jnjn N n D 2 Re

1 X nD1

nn N n :

(4.3)

4.1 Lecture IX. The action of quasisymmetric homeomorphisms on the Hilbert space 57

In other words, V is a Kähler Hilbert space. The complexification V C D H01=2 .S 1 ; C/ of the space V is the complex Hilbert space, consisting of functions f 2 L2 .S 1 ; C/ with Fourier decompositions of the form X f .z/ D fn z n ; z D e i ; n¤0

and finite Sobolev norm kf k21=2 D

X

jnjjfn j2 < 1:

n¤0

The Riemannian metric g 0 extends to V C as a Hermitian metric X h; i D jnjn N n : n¤0

We also extend the form ! and complex structure J 0 complex linearly to V C . Then the space V C will decompose into the direct sum of subspaces V C D WC ˚ W

(4.4)

where W˙ is the .i /-eigenspace of the linear operator J 0 W V C ! V C . In other words, ˚ ˚



P P n n W D  2 V C W .z/ D 1 WC D  2 V C W .z/ D 1 nD1 n z ; nD1 n z : The subspaces W˙ are isotropic with respect to the symplectic form !, i.e. !.; / D 0 if simultaneously ; 2 WC or ; 2 W . The decomposition (4.4) is the orthogonal direct sum with respect to the Hermitian inner product h; i. In terms of decomposition (4.4) this inner product is given by the formula h; i D i !.C ; N C /  i !. ; N  / where ˙ (resp. ˙ ) denotes the projection of  2 V C (resp. 2 V C ) onto W˙ . 4.1.2 Definition in terms of harmonic functions. Denote by D the Dirichlet space consisting of harmonic functions h W  ! R in the disk , normalized by the condition h.0/ D 0 and having finite energy E.h/ D

1 2

Z jgrad h.z/j2 dxdy D

1 2

Z

ˇ ˇ2 ˇ ˇ2 ! ˇ @h ˇ ˇ ˇ ˇ ˇ C ˇ @h ˇ dxdy < 1: ˇ ˇ ˇ ˇ

@x

@y

58

4 Grassmann realization of the universal Teichmüller space

Proposition 3. The Poisson transform Z 2 1 Pf .z/ D P .; z/f ./d; 2 0

 D e i ;

where P .; z/ is the Poisson kernel in the disk , P .; z/ D

jj2  jzj2 ; j  zj2

establishes an isometric isomorphism P W V ! D between the Sobolev space V and the Dirichlet space D, provided with the norm khk2D WD E.h/: Idea of the proof. Let f .z/ D

X

fn z n

n¤0

be an arbitrary element of V C . Then its Poisson transform h D Pf is given by the formula 1 1 X X n fn z C fn zN n (4.5) h.z/ D nD1

nD1

where z D re , 0  r < 1. Its energy is equal to Z X 1 E.h/ D jgrad h.z/j2 dxdy D jnjjfn j2 D kf k21=2 : 2

i

n¤0

Conversely, the Dirichlet space D is the subspace of the Sobolev space H01 ./, consisting of functions from L2 ./, having the generalized derivatives of the 1st order in L2 ./ and equal to zero at z D 0. Such functions have a correctly defined trace on the circle S 1 with “smoothness loss of 1/2”, in other words, the trace of a function from the space H01 ./ belongs to the space H01=2 .S 1 /. (This fact, well known in the theory of Sobolev spaces, may be found, for example, in the book [31].)  4.1.3 The action of quasisymmetric homeomorphisms on the Sobolev space. Let f be an orientation-preserving homeomorphism of S 1 . We associate with it the operator Tf , acting by the formula Z 2 1 .Tf /.z/ D .f .z//  .f .e i //d; z D e i ; 2 0 on functions  2 V .

4.1 Lecture IX. The action of quasisymmetric homeomorphisms on the Hilbert space 59

Theorem 17 (Nag–Sullivan theorem). The operator Tf acts from the space V into itself if and only if f 2 QS.S 1 /. If a quasisymmetric homeomorphism f extends to a K-quasiconformal homeomorphism of the disk  then the operator norm of Tf does p not exceed K C K 1 . Proof. Sufficiency. Suppose that a homeomorphism f 2 QS.S 1 / and extends to a quasiconformal homeomorphism w of the disk . Let  be an arbitrary vector in V and h D P  is its harmonic extension inside . Then the boundary value of the function g WD h B w is correctly defined and equal to  B f . We show that  B f 2 V , more precisely, that the energy of the harmonic extension of the function  B f does not exceed   1 C k2 E .P . B f //  2 E.Pf / (4.6) 1  k2 where k is the quasiconformality constant of w, equal to the norm kk1 of the Beltrami differential  of the map w. This will imply, by Proposition 3, that the operator norm of Tf is less than r r p 1 1Ck 1 C k2 kTf k  2 D KC where K D : 2 1k K 1k It is sufficient to prove the inequality (4.6) for the map g D h B w since this will imply its validity for P . B f /. (Recall that the minimum of energy among all smooth maps with the given boundary values is attained precisely on harmonic maps.) Setting w D u C iv, we obtain the estimate 

@g @x

2



C

@g @y

2



2

@h @u

2



C

@h @v

2 



N 2 almost everywhere in : j@wj2 C j@wj

N  kj@wj almost everywhere in The quasiconformality of the map w means that j@wj . Hence, 

@g @x

2



@g C @y

2



2

@h @u

2



@h C @v

2 



1 C k2 Jac.w/ 1  k2

N 2 . By changing the variable in the integral for E.g/ we where Jac.w/ D j@wj2  j@wj obtain the required inequality 

E.g/  2



1 C k2 E.h/: 1  k2

It is well known that the Dirichlet integral has the property of conformal invariance. The given proof shows in fact that it also has the quasi-invariance property with respect to quasiconformal maps. Necessity. Using the conformal invariance of the Dirichlet integral, we can reduce our assertion to the case of the upper halfplane H . Then the Sobolev space H01=2 .S 1 ; R/

60

4 Grassmann realization of the universal Teichmüller space

will be replaced by the space H 1=2 .R/. We shall need the following Douglas formula, expressing the energy of a map f 2 H 1=2 .R/ through its finite-difference derivative f (this formula is proved, for example, in the book [2]): E.Pf / D kf k21=2 D

1 4 2

Z Z  R

R

f .x/  f .y/ xy

2

dxdy:

(4.7)

Suppose now that f is an orientation-preserving homeomorphism f W R ! R for which Tf1 W H 1=2 .R/ ! H 1=2 .R/ is a bounded operator with norm M . We choose the bump function 0 2 C01 .R/, 0  0  1, such that 0  1 on the interval Œ1; 1 and 0  0 outside the closed interval Œ2; 2. Fix x 2 R, t > 0 and introduce the notation I1 D Œx  t; x;

I2 D Œx; x C t :

Consider the shift of the function 0 , given by the formula 1 .s/ WD 0 .as C b/ where the constants a; b are chosen in such a way that the function 1  1 on I1 and 1  0 on Œx C t; 1/. By assumption 1 B f 1 2 H 1=2 .R/ and, due to the boundedness of the operator 1 Tf , we have the following estimate: kTf1 1 k D k1 B f 1 k1=2  M k1 k1=2 D M k0 k1=2 (the latter equality follows, for example, from the Douglas formula). Hence, M 2 k0 k21=2  k1 B f 1 k21=2 Z Z   1 1 B f 1 .r/  1 B f 1 .s/ 2 D drds 4 2 R R r s Z f .x/ Z 1 1  ds dr .r  s/2 f .xt/ f .xCt/   f .x/  f .x  t / D log 1 C : f .x C t /  f .x/ It implies that 1 f .x/  f .x  t /  2 2 f .x C t /  f .x/ e M k 0 k1=2  1 for all x 2 R, t > 0.

(4.8)

4.1 Lecture IX. The action of quasisymmetric homeomorphisms on the Hilbert space 61

In analogous way, we can choose the shift 2 of the function 0 so that 2  1 on I2 and 2  0 on .1; x  t  and prove the opposite inequality 2 2 f .x C t /  f .x/  e M k 0 k1=2  1: f .x/  f .x  t /

It follows from the two last inequalities that the map f satisfies the Beurling–Ahlfors condition (1.8) from Section 1.3.3, i.e. this homeomorphism is quasisymmetric.  4.1.4 The action of quasisymmetric homeomorphisms on symplectic and complex structures Theorem 18. The action of operators Tf W V ! V with f 2 QS.S 1 / on the Sobolev space V preserves the symplectic structure !, i.e. !.Tf ; Tf / D !.; / for any ; 2 V:

(4.9)

Moreover, the complex-linear extension of the operator Tf to the complexified space V C preserves the subspaces W˙ if and only if f 2 Möb.S 1 / and in this case Tf acts on W˙ as a unitary operator. Idea of the proof. We start by proving the first assertion of the theorem. It is evidently true for smooth homeomorphisms f . Indeed, for smooth vectors ; 2 C01 .S 1 ; R/ this assertion is reduced to the change of variable, given by the map f , in the formula (4.1) for the form !0 from Section 4.1.1. Hence, the relation (4.9) is fulfilled also for arbitrary vectors ; 2 V . Approximating an arbitrary quasisymmetric homeomorphism f by the smooth ones with the help of approximation theorem from [17], Sec. 7.4, we prove the validity of the relation (4.9) for any f 2 QS.S 1 /. Next, if the action of Tf on V C preserves the subspace WC then f extends to a holomorphic map F W  !  (why?). Since f is a homeomorphism the map F should be conformal, i.e. F 2 Möb./. Since the operator Tf preserves the symplectic form it should also preserve the Hermitian metric on WC , i.e. acts on WC by unitary transforms. An analogous argument applies to W .  We show now that the form ! on V itself is defined essentially in a unique way. Theorem 19. Assume that a continuous bilinear form !Q W V V ! R has the following invariance property with respect to fractional-linear transforms: !.T Q f ; Tf / D !.; Q / for any f 2 Möb.S 1 /; ; 2 V: Then !Q D ! for some 2 R. In particular, if such a form is not identically zero, it is automatically non-degenerate and invariant under the whole group of quasisymmetric homeomorphisms QS.S 1 /.

62

4 Grassmann realization of the universal Teichmüller space

Idea of the proof. Any bilinear continuous map ! W V V ! R determines a duality map S W V ! V 0 , defined by the formula S./.  / WD !.;  / for  2 V: The duality map S, associated with the original symplectic form !, is evidently nondegenerate. Denote by Sz the duality map, associated with the form !, Q and consider the intertwining operator A WD S 1 Sz W V ! V: We show that this operator commutes with all invertible linear operators in V , preserving the forms ! and !. Q Indeed, the operator A is defined by the equality !.; A / D !.; Q /;

; 2 V:

If we have an invertible operator T , preserving the forms ! and !, Q then the following relations take place !.T ; TA / D !.; A / D !.; Q / D !.T Q ; T / D !.T ; AT /: Since the operator T is invertible it follows that !.; TA / D !.; AT / for all  2 V: But the form ! is non-degenerate so TA D AT , as it was stated. We want to show that the intertwining operator A is scalar which will imply that Sz D S H) !Q D !. To prove that we use the Schur lemma from the representation theory. Note that the unitary representation of the group Möb.S 1 / D PSL.2; R/ in V C , given by the operators Tf with f 2 Möb.S 1 /, determines an irreducible unitary representation of the group SL.2; R/ in the spaces W˙ and these spaces are the only invariant subspaces of this representation in V C . (This fact follows from the general representation theory of the group SL.2; R/, cf. [20], Lemma 4.6.) As we have proved, the intertwining operator A must commute with all invertible operators, preserving the forms ! and !. Q In particular, it should commute with all operators Tf with f 2 Möb.S 1 /. Since the operators Tf with f 2 Möb.S 1 / preserve the subspaces W˙ , the operator A may send the subspace WC only to WC or to W . If the first possibility is realized then A should commute with all operators Tf W WC ! WC of the irreducible unitary representation of the group SL.2; R/ in WC and so is scalar, i.e. A D I where 2 R due to the reality of the operator A. The second possibility cannot be realized since in this case it would be possible to replace the operator A by N mapping WC to WC . As we have proved, such an the complex conjugate operator A, N operator A must be scalar, in other words, it should be equal to I with 2 R, i.e. coincide with A, which is impossible. 

4.1 Lecture IX. The action of quasisymmetric homeomorphisms on the Hilbert space 63

Brief content of Lecture IX The Sobolev space of half-differentiable functions V D H01=2 .S 1 ; R/; consisting of functions f 2 L2 .S 1 ; R/ with Fourier decompositions of the form X fn z n ; fNn D fn ; f .z/ D n¤0

with finite norm kf k21=2 D

X

jnjjfn j2 :

n¤0

This is a Kähler Hilbert space, having a symplectic form ! and a complex structure J 0 , which are compatible with each other and generate a Riemannian metric g 0 . The complexification V C D H01=2 .S 1 ; C/ is decomposed into the direct sum V C D WC ˚ W of .i/-eigenspaces of the operator J 0 which is orthogonal with respect to the inner product h; i, given by the Hermitian extension of the metric g 0 to V C . The Sobolev space V may be identified with the Dirichlet space D, consisting of harmonic functions h in the disk , normalized by the condition h.0/ D 0 and having finite energy Z 1 E.h/ D jgrad h.z/j2 dxdy < 1: 2

An isometric isomorphism V ! D is established via the Poisson transform. Nag–Sullivan theorem: 1. The transformation f 7! Tf  WD  B f  average. B f / 1 acts p from the space V into itself () f 2 QS.S / and in this case kTf k  K C K 1 where K is the maximal dilatation of f .

2. For f 2 QS.S 1 / the operator Tf acts by symplectic transforms; the complexlinear extension of the operator Tf to V C preserves the subspaces W˙ () f 2 Möb.S 1 /. 3. If f 2 Möb.S 1 / then Tf acts on W˙ by unitary transforms. Theorem: The symplectic form ! on V is essentially uniquely defined, namely, if !Q is another continuous bilinear form on V , invariant under transformations Tf with f 2 Möb.S 1 / then !Q D ! for some 2 R.

64

4 Grassmann realization of the universal Teichmüller space

4.2 Lecture X. Grassmann realization of the space T 4.2.1 Embedding of the space T into the Siegel disk. The Nag–Sullivan theorem (Section 4.1.3) and Theorem 18 (Section 4.1.4) imply that there is an embedding T D QS.S 1 /=Möb.S 1 / ! Sp.V /=U.WC /

(4.10)

where Sp.V / denotes the symplectic group of the space V , consisting of bounded linear operators on V , preserving the symplectic form !, and U.WC / is its subgroup, consisting of unitary operators, i.e. operators, having complex-linear extensions to V C which preserve the subspace WC (hence, also W ). Let us describe these groups in more detail. In terms of the decomposition (4.4) from Section 4.1.1, V C D WC ˚ W ; any linear operator A W V C ! V C may be written down in the block form     a b a W WC ! WC b W W ! WC : AD D c W WC ! W d W W ! W c d In particular, linear operators on V C , obtained by the complex-linear extension of operators A W V ! V , have the block representations of the form   a b AD N b aN SC . where we identify the space W with the complex conjugate space W A linear operator A W V ! V belongs to the symplectic group Sp.V / if it preserves the symplectic form !. This condition is equivalent to the following relations for block components of A (check it!):   a b AD N 2 Sp.V / () aN t a  b t bN D 1; aN t b D b t aN (4.11) b aN where at , b t denote the transposed operators at W WC0 ! WC0 () at W W ! W ; b t W WC0 ! W0 () b t W W ! WC where the space WC0 , dual to the space WC , is identified with the space W by means of the inner product .; / on V C , given by the complex-linear extension to V C of the Riemannian metric g 0 . Using the equality A1 A D I , we can obtain also the dual relations aaN t  b bN t D 1; abN t D bat : (4.12)

4.2 Lecture X. Grassmann realization of the space T

65

The unitary group U.WC / is embedded into the symplectic group Sp.V / as the subgroup of block-diagonal matrices of the form   a 0 AD : 0 aN We return to the map (4.10). The space Sp.V /=U.WC / on the right-hand side of (4.10) may be identified with the space J.V / of complex structures on the space V C , compatible with the symplectic form !. Indeed, any complex structure J of this type determines the decomposition S VC DW ˚W

(4.13)

into the direct sum of .i /-eigenspaces of the operator J , isotropic with respect to !. Conversely, any decomposition of the form (4.13) of the space V C into the direct sum of subspaces, isotropic with respect to !, determines a complex structure J on V C , S and compatible with !. Thus, the group Sp.V / acts equal to iI on W and CiI on W transitively on the space J.V / of complex structures J on V , compatible with !. To obtain a homogeneous representation for the space J.V / it is necessary to take the quotient of the group Sp.V / over its subgroup, consisting of the transformations, preserving the original complex structure J 0 or, in other words, preserving the subspaces W˙ . This subgroup consists precisely of the unitary transformations from the group U.WC /, which implies that J.V / D Sp.V /=U.WC /: The space J.V / admits an interpretation as an infinite-dimensional Siegel disk. By definition, the Siegel disk D consists of bounded linear operators of the form x < I g: D D fZ W WC ! W is a bounded linear symmetric operator such that ZZ x < I is equivalent The symmetricity of Z means that Z t D Z, and the condition ZZ x to the fact that the symmetric operator I  ZZ is positively definite. In order to identify the space J.V / with the Siegel disk D consider the action of the group Sp.V / on D, given by the operator fractional-linear transformations of the form   a b N Sp.V / 3 A D N W Z 7! .aZ N C b/.bZ C a/1 : b aN It is easy to check that the assignment of such a fractional linear transformation of the Siegel disk D to an operator A 2 Sp.V / establishes a one-to-one map J.V / D Sp.V /=U.WC / ! D:

66

4 Grassmann realization of the universal Teichmüller space

The Siegel disk D is naturally embedded into the Grassmannian Grb .V C / of the Hilbert space V C , consisting of the closed subspaces W  V C , which are obtained from WC by the action of bounded linear operators. This embedding is given by the map D 3 Z 7! the graph of the map Z W WC ! W : The Grassmannian Grb .V C / is a complex Banach manifold (cf. [27], Sec. 5.1), and the composite map T D QS.S 1 /=Möb.S 1 / ! Sp.V /=U.WC / D D ! Grb .V C / is an equivariant holomorphic embedding of complex Banach manifolds. (This fact is proved in the paper [20].) 4.2.2 Grassmann realization of the space of normalized diffeomorphisms. The constructed embedding T ,! Grb .V C / generates an embedding of the space of normalized diffeomorphisms  D DiffC .S 1 /=Möb.S 1 /  T into the “regular part” of the Grassmannian Grb .V C /, coinciding with the Hilbert– Schmidt Grassmannian GrHS .V / which is defined in the following way. Definition 17. The Hilbert–Schmidt Grassmannian GrHS .V / consists of closed subspaces W  V C such that the orthogonal projection C W W ! WC is a Fredholm operator, and the orthogonal projection  W W ! W is a Hilbert–Schmidt operator. Recall that a linear operator T W H1 ! H2 , acting from a Hilbert space H1 into a Hilbert space H2 , is called Fredholm if it has finite-dimensional kernel and cokernel. Such an operator is invertible modulo compact operators, i.e. there exists a linear operator S W H2 ! H1 such that the operators IH1  ST

and IH2  T S

are compact. An operator T is called the Hilbert–Schmidt operator (HS-operator) if for some orthonormal basis fei g in the space H1 the series X 2 kT ei kH <1 2 is convergent. If this condition is fulfilled for some orthonormal basis fei g in H1 , it is also satisfied for any orthonormal basis in H1 . The Hilbert–Schmidt operators form a complex Hilbert space HS.H1 ; H2 / with the norm kT k WD

X

2 kT ei kH 2

1=2

:

4.2 Lecture X. Grassmann realization of the space T

67

Returning to the definition of the Hilbert–Schmidt Grassmannian, one can say that GrHS .V / consists of closed subspaces W  V C which differ “little” from the subspace WC in the sense that the projection C W W ! WC is “almost” invertible, while the projection  W W ! W is “small”. The Grassmannian GrHS .V / is a Kähler Hilbert manifold, having as a local model the Hilbert space HS.WC ; W / of Hilbert–Schmidt operators. We now introduce the Hilbert–Schmidt symplectic group SpHS .V / which consists of transformations   a b AD N 2 Sp.V / b aN for which b is a Hilbert–Schmidt operator. The unitary group U.WC / is contained in SpHS .V / as the subgroup of block-diagonal matrices. The constructed embedding T ,! J.V / induces the embedding  D DiffC .S 1 /=Möb.S 1 / ,! SpHS .V /=U.WC /: Problem 7. Prove that the image of the group DiffC .S 1 / in Sp.V / under the embedding QS.S 1 / ,! Sp.V / belongs, in fact, to the subgroup SpHS .V /. The space JHS .V / WD SpHS .V /=U.WC / is identified, as in Section 4.2.1, with some space of complex structures on V C , compatible with the symplectic form !. We shall call the complex structures from JHS .V / the Hilbert–Schmidt complex structures. Similar to Section 4.2.1, the space JHS .V / admits a realization in the form of the Hilbert–Schmidt Siegel disk, defined as x < I g: DHS D fZ W WC ! W is a symmetric Hilbert–Schmidt operator with ZZ By analogy with Section 4.2.1, one can prove that the Hilbert–Schmidt Siegel disk DHS can be embedded into the Hilbert–Schmidt Grassmannian GrHS .V / so that the composite map  D DiffC .S 1 /=Möb.S 1 / ,! JHS .V / D SpHS .V /=U.WC / D DHS ,! GrHS .V / is an equivariant holomorphic embedding of the Frechet complex space  into the complex Hilbert manifold GrHS .V / (cf. [19]). Brief content of Lecture X The Nag–Sullivan theorem implies that T D QS.S 1 /=Möb.S 1 / ,! Sp.V /=U.WC / D J.V /: The space J.V / of complex structures on the Sobolev space V , compatible with the symplectic form !, is identified with the Siegel disk x < I g: D D fZ W WC ! W is a symmetric bounded linear operator with ZZ

68

4 Grassmann realization of the universal Teichmüller space

This identification is established with the help of the map, associating with an operator   a b Sp.V / 3 A N b aN the fractional-linear transformation of the Siegel disk of the form N Z 7! .aZ N C b/.bZ C a/1 : The composite map

T ,! J D D ,! Grb .V C /

of the universal Teichmüller space T into the Grassmannian Grb .V C / of the Sobolev space V C is a holomorphic embedding of complex Banach manifolds. The restriction of this embedding to the space  of normalized diffeomorphisms generates an embedding  D DiffC .S 1 /=Möb.S 1 / ,! SpHS .V /=U.WC / D JHS .V / where SpHS .V / D fA 2 Sp.V / W b is a Hilbert–Schmidt operatorg is the Hilbert–Schmidt symplectic group, and JHS .V / is the space of Hilbert–Schmidt complex structures on V , compatible with the symplectic form !. The latter space is identified with the Hilbert–Schmidt Siegel disk x < I g: DHS D fZ W WC ! W is a symmetric Hilbert–Schmidt operator with ZZ The composite map  ,! JHS .V / D DHS ,! GrHS .V C / of the space  into the Hilbert–Schmidt Grassmannian GrHS .V C / of the Hilbert space V C is a holomorphic embedding of the complex Frechet space  into the complex Hilbert manifold GrHS .V /.

5 Quantization of the space of normalized diffeomorphisms

This chapter is devoted to the quantization of the space  of normalized diffeomorphisms. It starts with Lecture 5.1 in which we give a definition of classical systems and their quantization, according to Dirac. In Section 5.1.3 the Dirac quantization problem is specialized to the case of the space . We first construct the quantization of the extended system having phase manifold given by the space JHS .V / of Hilbert–Schmidt complex structures on the Sobolev space V . The role of the quantization space is played in this case by the Fock space, associated with the space V and a complex structure J on it (Section 5.2.1). The Heisenberg algebra heis.V / has an irreducible unitary Heisenberg representation in this space (Section 5.2.2). A key role in the quantization of the space JHS .V / is played by the Shale–Berezin theorem given in Section 5.2.3. It asserts that the Heisenberg representations in the Fock spaces associated with two different complex structures on the space V are unitary equivalent if and only if these structures may be obtained from each other by Hilbert–Schmidt symplectic transformations. This theorem implies that the Fock bundle, unifying the Fock spaces, associated with different Hilbert–Schmidt complex structures on V , is a holomorphic Hilbert bundle over JHS .V /, provided with a projective action of the Hilbert–Schmidt symplectic group, covering the action of this group on the base of the bundle. The infinitesimal version of this action yields a projective unitary representation of the Hilbert–Schmidt symplectic algebra in the Fock space, constructed in Section 5.2.4. This representation determines the Dirac quantization of the space JHS .V / and its restriction to  gives the quantization of the original space . The quantization of  may be also constructed directly, using the representation of the Virasoro algebra discussed in Section 5.2.5.

5.1 Lecture XI. Quantization of classical systems by Dirac 5.1.1 Classical systems. A finite-dimensional classical system is given by the pair .M; A/, consisting of the phase space M and the algebra of observables A. The phase space M is a smooth symplectic manifold of an even dimension 2n with the symplectic form !. Locally, it is isomorphic to the standard model M0 WD .R2n ; !0 / where !0 is the standard symplectic form on R2n , given in the canonical

70

5 Quantization of space of diffeomorphisms

coordinates .pi ; qi /, i D 1; : : : ; n, on R2n by the formula !0 D

n X

dpi ^ dqi :

iD1

The algebra of observables A is an arbitrary Lie subalgebra in the Lie algebra C 1 .M; R/ of smooth real-valued functions on the phase space M with respect to the Poisson bracket, determined by the symplectic form !. In particular, A may coincide with the whole Poisson algebra C 1 .M; R/. In the case of the standard model M0 D .R2n ; !0 / one can take as an algebra of observables the Heisenberg algebra heis.R2n / which is generated by the coordinate functions pi ; qi , i D 1; : : : ; n, and 1, satisfying the commutation relations: fpi ; pj g D fqi ; qj g D 0; fpi ; qj g D ıij for i; j D 1; : : : ; n: The algebras of observables arise usually in the following way. Let  be a Lie group, acting on the simply connected phase manifold M by symplectic transformations. Then its Lie algebra Lie./ may be considered as a subalgebra of the Lie algebra of Hamiltonian vector fields Xf on M , generated by smooth functions f 2 C 1 .M; R/. In this case we can take for the algebra of observables the Lie algebra ham./, consisting of functions f for which Xf 2 Lie./, provided with the Poisson bracket as a Lie bracket. 5.1.2 Quantization of classical systems by Dirac. Let .M; A/ be a classical system. The quantization of this system by Dirac is an irreducible linear representation r W A ! End H of observables from A by selfadjoint linear operators, acting in a complex Hilbert space H , called the quantization space. It is required that r .ff; gg/ D

1 1 Œr.f /; r.g/ D .r.f /r.g/  r.g/r.f // i i

(5.1)

for any f; g 2 A and r.1/ D I . The quantization operators r.f /, arising in concrete examples, are usually unbounded so it is necessary to require that all of them should be defined on a common definition domain which is dense in H . It is often more convenient to deal with the complexified algebras of observables AC or, more generally, with the involutive complex algebras of observables AC , provided with an involution. In this case the Dirac quantization of an algebra of observables AC will be given by an irreducible linear representation r W AC ! End H by closed linear operators in H , satisfying, apart from the condition (5.1) and normalization r.1/ D I ,

5.1 Lecture XI. Quantization of classical systems by Dirac

71

also to the conjugation rule: the involution in AC corresponds under the action of r to the Hermitian conjugation. We shall apply the given definition of Dirac quantization to the infinite-dimensional classical systems in which both the phase spaces and the algebras of observables are infinite-dimensional. For the infinite-dimensional Lie algebras A it is more natural to look for projective, rather than ordinary representations. If we succeed in finding such a representation for a given algebra of observables A, it will mean that we have z of the original system .M; A/ constructed the quantization of the extension .M; A/ z where A is an appropriate central extension of the algebra A, determined by the cocycle of the projective representation. (On projective representations and central extensions cf., e.g., [27], Ch. 4.) 5.1.3 Quantization of the space : statement of the problem. In the case of the space of normalized diffeomorphisms  D DiffC .S 1 /=Möb.S 1 / the role of an infinite-dimensional classical system is played by the pair   ; Vect.S 1 / where  is the phase space of the system, and Vect.S 1 / is the algebra of observables which is the Lie algebra of the Lie group DiffC .S 1 /. This algebra coincides with the Lie algebra of smooth vector fields on S 1 . We shall construct the quantization of this system by extending it first to a system, associated with the Sobolev space V . For that we shall use the embedding  ,! JHS .V / D SpHS .V /=U.WC /; constructed in Section 4.2.2. Under this map the group DiffC .S 1 / is embedded into the Hilbert–Schmidt symplectic group SpHS .V /. Then we take for the extended classical system the pair .JHS .V / D DHS ; spHS .V // where spHS .V / is the Lie algebra of the Hilbert–Schmidt symplectic group SpHS .V /. Brief content of Lecture XI Classical system: .M; A/ where M is the phase space, A is the algebra of observables on M . The standard model: M D R2n , A D heis.R2n /. If  is a Lie group of symplectomorphisms on a simply connected symplectic manifold M then one can associate with it a classical model with the phase space M and algebra of observables A D Lie  where Lie  is the Lie algebra of the Lie group . The Dirac quantization of a classical system .M; A/ is an irreducible linear representation r of observables from the algebra A by selfadjoint linear operators, acting in a

72

5 Quantization of space of diffeomorphisms

complex Hilbert quantization space H , under which the Poisson bracket of observables is sent to the commutator of the corresponding operators: 1 Œr.f /; r.g/: i In the case of the space of normalized diffeomorphisms ff; gg 7!

 D DiffC .S 1 /=Möb.S 1 / the role of an infinite-dimensional classical system is played by the pair   ; Vect.S 1 / with the phase space  and algebra of observables Vect.S 1 /, coinciding with the Lie algebra of smooth vector fields on S 1 . This classical system is embedded into the extended system .JHS .V /; spHS .V // where spHS .V / is the Lie algebra of the Hilbert–Schmidt symplectic group SpHS .V /.

5.2 Lecture XII. Quantization of the extended system 5.2.1 The Fock space. To start the construction of the quantization of the extended system .JHS .V /; spHS .V //, we should indicate, first of all, what is the quantization space H in which the representation of the algebra of observables spHS .V / will be realized. The role of this space in the considered case is played by the Fock space associated with the Sobolev space V . To define this space, we fix some complex structure J 2 J.V / compatible with the symplectic form !. This structure generates the decomposition of the complexified space V C into the direct sum S VC DW ˚W of .i/-eigenspaces of operator J . This decomposition is orthogonal with respect to the Hermitian inner product on V C , generated by J and !: hz; wiJ WD !.z; J w/: The Fock space F .V C ; J / is the completion of the algebra of symmetric polynomials in variables z 2 W with respect to the norm generated by the inner product h; iJ . In more detail, denote by S.W / the algebra of symmetric polynomials in variables z 2 W and introduce on it the inner product, generated by the inner product h; iJ . It is given on monomials of equal degrees by the formula X hz1 ˝    ˝ zn ; z10 ˝    ˝ zn0 iJ WD hz1 ; zi01 iJ : : : hzn ; zi0n iJ fi1 ;:::;in g

73

5.2 Lecture XII. Quantization of the extended system

where the summation is taken over all permutations fi1 ; : : : ; in g of the set f1; : : : ; ng (the inner product of monomials of different degrees is set to zero). This inner product on monomials is then extended by linearity to the algebra S.W /. Definition 18. The Fock space FJ D F .V C ; J / is the completion of the algebra S.W / with respect to the norm h; iJ . If fwn g1 nD1 is an orthonormal basis of the space W then one can take for the orthonormal basis of the Fock space FJ the monomials of the form 1 PK .z/ D p hz; w1 ikJ1 : : : hz; wn ikJn ; kŠ

z 2 W;

(5.2)

where K D .k1 ; : : : ; kn ; 0; : : :/ is a finite collection of natural numbers ki 2 N, kŠ D k1 Š : : : kn Š. Thus, the Fock space coincides with the completion of the direct sum FJ D

1 M

Sk .W /

kD0

where Sk .W / is the subspace of homogeneous polynomials of degree k in S.W /. 5.2.2 Heisenberg representation Definition 19. The Heisenberg algebra heis.V / of the Hilbert space V is the central extension of the Abelian Lie algebra V generated by the coordinate functions. In other words, as a vector space, this algebra coincides with heis.V / D V ˚ R and is provided with the Lie bracket of the form Œ.x; s/; .y; t / WD .0; !.x; y// ;

x; y 2 V; s; t 2 R:

We shall construct now an irreducible representation of the Heisenberg algebra heis.V / in the Fock space FJ . Note, first of all, that the elements of the algebra S.W / S by identifying z 2 W may be considered as holomorphic functions on the space W with the holomorphic function S 3w W x 7! hz; wiJ

S: on W

Accordingly, the space FJ may be considered as a space of functions, holomorphic S. on W Taking into account this identification, we can define the Heisenberg representation rJ of the Heisenberg algebra heis.V / in the Fock space FJ by the formula x D @v f .w/ x C hv; wiJ f .w/ x V 3 v 7! rJ .v/f .w/

(5.3)

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5 Quantization of space of diffeomorphisms

where @v is the operator of differentiation in the direction of vector v. Extending rJ to the complexified algebra heisC .V / by the same formula (5.3), we obtain S; rJ .z/f N .w/ x D @zN f .w/ x for zN 2 W x D hz; wiJ f .w/ x for z 2 W: rJ .z/f .w/ It is convenient to describe the Heisenberg representation in terms of the creation and annihilation operators on the space FJ , given by the formulas aJ .v/ D

rJ .v/ C i rJ .J v/ ; 2

aJ .v/ D

rJ .v/  i rJ .J v/ 2

where v 2 V C . It implies that aJ .z/f .w/ x D hz; wiJ f .w/ x for z 2 W; S: aJ .z/f N .w/ x D @zN f .w/ x for zN 2 W Choosing an orthonormal basis fwn g1 nD1 in the space W , we introduce the operators an WD a .wn /;

an WD a.w xn /

for n D 1; 2; : : : :

These operators satisfy the commutation relations of the form  ; an  D 0; Œam ; an  D Œam

 Œam ; an  D ımn I

for m; n 2 N:

(5.4)

Definition 20. A vector fJ 2 FJ n f0g is called the vacuum if it is annihilated by all annihilation operators, i.e. an fJ D 0

for all n D 1; 2; : : : :

(5.5)

Such a vector is uniquely determined by the representation rJ up to a multiplicative constant. In the case of the original Fock space F0 D F .V; J 0 / we take for the vacuum f0  1. Acting on the vacuum fJ by the creation operators an , we shall obtain the set of vectors in FJ of the form .a1 /k1 : : : .an /kn fJ with the closed linear hull being equal to the whole space FJ , which implies the irreducibility of the representation rJ . Note that the monomials PK .z/, given by the formula (5.2) from Section 5.2.1, taken for the orthonormal basis of the space FJ , are constructed precisely by this method. We show that any irreducible representation rQ of the Heisenberg algebra heis.V / with a vacuum is equivalent to the original representation r0 in the Fock space F0 D F .V; J 0 /. Indeed, the vectors of the form .aQ 1 /k1 : : : .aQ n /kn fQ, obtained from the vacuum fQ of the representation rQ by the action of the corresponding creation operators aQ n , are linearly independent and generate an invariant subspace in the space Fz of the representation r. Q Due to the irreducibility, this subspace should coincide with the whole z space F . Consider now the map F0 ! Fz , associating with a polynomial P .z1 ; : : : ; zn /

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from the algebra S.W / the vector of the form P .aQ 1 ; : : : ; aQ n /fQ in the space Fz . This map may be made unitary if we introduce on Fz the Hermitian inner product, in which the vectors p1 .aQ 1 /k1 : : : .aQ n /kn fQ form an orthonormal basis. Then the constructed kŠ

map F0 ! Fz will define a unitary operator, intertwining the representation rQ with the original representation r0 . 5.2.3 Shale–Berezin theorem. We want to construct a unitary operator UJ W F0 ! FJ , intertwining the Heisenberg representations r0 in the space F0 and rJ in the space FJ . Theorem 20 (Shale–Berezin theorem (cf. [29], [3]). Suppose that a complex structure J 2 J.V / is obtained from the complex structure J 0 by the action of an element A 2 Sp.V /. Then the representations r0 in the space F0 and rJ in the space FJ are unitary equivalent if and only if A 2 SpHS .V /. In other words, if the latter condition is satisfied then there exists a unitary intertwining operator UJ W F0 ! FJ such that rJ D UJ B r0 B UJ1 : Idea of the proof (for a complete proof see Sect. 4.3 in Part I, Ch. II, of the book [3]). To construct an intertwining operator UJ it is sufficient, according to the argument given at the end of Section 5.2.2, to construct a vacuum in the space FJ . We can try to find this vacuum from its decomposition in the basis of the space F0 , formed by the vectors p1 .a1 /k1 : : : .an /kn f0 , by substituting the obtained series into the relations (5.5). kŠ Then the required vacuum fJ will be given by the formula 1



fJ D ce  2 aJ .a

1 b/a J

f0

(5.6)

 for A D

 a b . The coefficient c here is equal to bN aN cD

 .det aaN t /1=4

where  is a complex number with modulus 1. Note that from the description of the group Sp.V / given by the relations (4.11) from Section 4.2.1, it follows that the operator a is invertible. Moreover, the vector fJ given by the formula (5.6) belongs to the space FJ if and only if a1 b is a Hilbert–Schmidt operator () b is a Hilbert–Schmidt operator, i.e. A 2 SpHS .V /. In this case the operator (cf. the formula (4.12)) aaN t D 1 C b bN t has the form “1 + trace class operator”, so its determinant is correctly defined. The undetermined coefficient  arises because the vacuum fJ is determined only up to a multiplicative constant with modulus 1.

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5 Quantization of space of diffeomorphisms

Having the formula (5.6) for the vacuum fJ , one can find also an explicit representation for the intertwining operator UJ , which may be found in the aforementioned book [3], Part I, Ch. II, Sec. 4.5. This operator, as also the vacuum, is uniquely defined up to a multiplicative constant with modulus 1.  We unify now all the Fock spaces FJ with J 2 JHS .V / into a single Fock bundle [ FJ ! JHS .V / D DHS : F D J 2JHS .V /

Proposition 4. The Fock bundle F ! JHS .V / is a Hermitian holomorphic Hilbert bundle over the Siegel disk DHS . There is a unitary projective action of the group SpHS .V /, covering the natural action of this group on JHS .V / D DHS . The holomorphicity of the Fock bundle is established in the same way as that of the determinant bundle over the Hilbert–Schmidt Grassmannian GrHS .V / (cf. [27], Sec. 5.3). Since the Siegel disk DHS is a contractible (and even convex) set, this bundle is trivial. Moreover, the action of the group SpHS .V /, determined by the Shale–Berezin theorem, yields its explicit trivialization. 5.2.4 Representation of the symplectic Hilbert–Schmidt algebra. The infinitesimal version of the action of the Hilbert–Schmidt symplectic group SpHS .V / in the Fock space yields a projective representation of its Lie algebra spHS .V / in the fibre F0 D F .V C ; J 0 / of the Fock bundle at the point J 0 . An explicit construction of this representation is given in the paper [25]. The symplectic Lie algebra spHS .V / consists of bounded linear operators A, acting in the space V C and having the block representations of the form   ˛ ˇ AD N ˇ ˛N where ˛ is a bounded skew-Hermitian operator, ˇ is a symmetric Hilbert–Schmidt operator. The complexified Lie algebra spHS .V /C consists of operators of the form   ˛ ˇ AD N ˛ t where ˛ is a bounded operator, while ˇ and N are symmetric Hilbert–Schmidt operators. The projective representation of the complexified symplectic Lie algebra spHS .V /C in the space F0 is given by the formula   1 1 ˛ ˇ (5.7) 7! .A/ D D˛ C Mˇ C M  : spHS .V /C 3 A D t N ˛ 2 2 Here, D˛ is the differentiation operator, generated by the operator ˛ W WC ! WC and defined by the formula D˛ f .w/ x D h˛w; @w if .w/: x

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77

SC ! WC has the form The operator Mˇ generated by the operator ˇ W W D W N wif .w/; Mˇ f .w/ x D hˇw; x and the operator M  , conjugate to M , acts by the formula x D h @w ; @w if .w/: x M  f .w/ Theorem 21 (Segal theorem [25]). The formula (5.7) determines a unitary projective representation of the Lie algebra spHS .V /C in the Fock space F0 with the cocycle Œ.A1 /; .A2 /  .ŒA1 ; A2 / D

1 tr.N2 ˇ1  N1 ˇ2 /I: 2

(5.8)

This representation intertwines with the Heisenberg representation r0 of the Heisenberg algebra heis.V / in the space F0 . The assertion that the constructed representation of the Hilbert–Schmidt symplectic algebra intertwines with the representation r0 follows from the fact that this representation is the infinitesimal version of the projective action of the Hilbert–Schmidt symplectic group on the Fock space, intertwining different representations of the Heisenberg algebra. The projective representation of the Lie algebra spHS .V / determines the Dirac quantization of the extended system   JHS .V / D DHS ; spHS .V /

B

B

where spHS .V / is the central extension of the Lie algebra spHS .V /, determined by the cocycle (5.8). We have constructed simultaneously the Dirac quantization of one more classical system, closely related to the string theory, namely, the system .V; A/; having the phase space given by the Sobolev space V D H01=2 .S 1 ; R/ and algebra of observables A equal to the semi-direct sum A D heis.V / Ì spHS .V /: This algebra of observables may be considered as an infinite-dimensional analogue of the Poincaré algebra of the Minkowski space. Recall that the Poincaré algebra is the semi-direct sum of the algebra of translations and the algebra of hyperbolic rotations of the Minkowski space. In the case of the Sobolev space V the role of the algebra of translations is played by the Heisenberg algebra, while the role of the algebra of rotations is played by the symplectic Lie algebra spHS .V /. In the case of the Minkowski space transformations from the algebra of translations depend linearly on

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5 Quantization of space of diffeomorphisms

the coordinates, while transformations from the algebra of rotations depend on them quadratically. This behavior is preserved also in the infinite-dimensional case – the Heisenberg representation is linear with respect to the variables w x and @wx , while the representation of the Lie algebra spHS .V / depends on them quadratically. Let us consider the relation between the constructed representation of the Lie algebra spHS .V / and the invariant connections on the Fock bundle. Note that the symplectic Lie algebra spHS .V / admits a decomposition into the direct sum spHS .V / D u.WC / ˚ m where u.WC / is the Lie algebra of the unitary group U.WC / identified with the set of matrices of the form   ˛ 0 0 ˛ t where ˛ W WC ! WC is a bounded skew-symmetric operator. The linear subspace m consists of matrices of the form   0 ˇ ˇN 0 where ˇ W W ! WC is a symmetric Hilbert–Schmidt operator. This subspace may be identified with the tangent space T0 DHS . It is invariant under the adjoint action of the group U.WC / on the Lie algebra spHS .V /. In accordance with the general theory of invariant connections (cf., e.g., [15], Vol. I, Ch. II, Sec. 11) there exists a bijective correspondence between projective unitary representations of the Lie algebra spHS .V / in the space F0 and unitary SpHS .V /-invariant projectively-flat connections on the Fock bundle F . This correspondence is established in the following way: the connection form is given by the restriction of the projective representation to the subspace m. Then the curvature of the connection will coincide with the cocycle of the representation. In more detail, any two of the three objects in Figure 1 below determine the third one. For example, the connection form in the fibre F0 is determined by the representation of spHS .V / in F0 , restricted to the subspace m, and then transported to other fibres of F by the action of the group SpHS .V /. The representation and connection form define the action of the group SpHS .V / by the integration of the representation with the help of connection. The representation is reconstructed from the connection form, invariant under the group action, by subtracting the flat connection from it. 5.2.5 Quantization of the space of normalized diffeomorphisms . The restriction of the Fock bundle F ! DHS , constructed in Section 5.2.3, to the submanifold  D DiffC .S 1 /=Möb.S 1 /  JHS .V / D DHS yields the Fock bundle F WD

[ J 2

FJ ! 

5.2 Lecture XII. Quantization of the extended system

79

over the space . projective representation of spHS .V / in F0

projective action of SpHS .V / on F

form of invariant projectively flat connection on F

Figure 1

Proposition 5. The Fock bundle F !  is a Hermitian holomorphic Hilbert bundle over the space . There is a projective unitary action of the diffeomorphism group DiffC .S 1 / on this bundle, covering the natural action of this group on . The Fock bundle F !  is trivial, since the space  is contractible (try to prove it by yourself!). The action of the group DiffC .S 1 / on the bundle F is given by the restriction of the SpHS .V /-action on the Fock bundle F ! DHS , constructed in Section 5.2.3. However, this action may be also constructed explicitly, as in the paper [13]. The infinitesimal version of the action of the group DiffC .S 1 / on the Fock bundle F is a projective representation of the Lie algebra Vect.S 1 / of this group in the Fock space F0 . It is convenient to describe this representation, called the Virasoro representation, in terms of the creation and annihilation operators an ; an on the space F0 , introduced in Section 5.2.2. We supplement their definition with the following notation a0 D I; 2 R;

and

an WD nan ; n 2 N;

so that for the obtained operators the following commutation relations hold Œam ; an  D mım;n I

for m; n 2 Z:

The Virasoro representation of the Lie algebra VectC .S 1 / is generated by the Virasoro operators Ln which are the images of the basis elements en of the algebra VectC .S 1 /. These operators are given by the formula 1 1 X W ai aiCn W; Ln D 2 iD1

n 2 Z;

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5 Quantization of space of diffeomorphisms

where the notation W W means the normal ordering, defined by the rule ´ ai aj for i  j; W ai aj WD aj ai for i > j: These operators may be also written down without normal ordering in the form 8P < i>n=2 ai aiCn for odd n; Ln D a2 P : n=2 C i>n=2 ai aiCn for even n: 2 In particular, the energy operator L0 has the form L0 D

2 X ai ai : C 2 i>0

Because of the normal ordering, the application of the operator Ln to any polynomial P from the algebra S.WC / will involve only a finite number of non-vanishing terms in the infinite series, defining Ln P so the action of operators Ln is correctly defined on the algebra S.WC / and may be extended to the whole Fock space F0 D S.WC / by completion. The operators Ln generate a unitary projective representation of the Lie algebra Vect.S 1 / in the Fock space F0 since they satisfy the following commutation relations

2

ŒLm ; Ln  D .m  n/LmCn C ım;n

m3  m 12

(5.9)

(cf. [14], Sec. 2.3). Remark 5 (cf. [14]). This representation is irreducible for generic values of the parameter . The constructed projective representation of the Lie algebra Vect.S 1 / in the Fock space F0 determines the Dirac quantization of the system .; vir/ where vir is the central extension of the Lie algebra Vect.S 1 /. This extension is called the Virasoro algebra and is determined by the cocycle of the representation (5.9). Note that the central extension of Vect.S 1 / is essentially uniquely defined (cf. [27], Sec. 10.1). Brief content of Lecture XII To start the quantization of the extended system .JHS .V /; spHS .V // ;

5.2 Lecture XII. Quantization of the extended system

81

we take for the quantization space the Fock space, associated with the Sobolev space V C. The Fock space, associated with the Sobolev space V C , provided with the complex structure J 2 J.V /, is defined in the following way. The complex structure J generates the decomposition of the complexified Sobolev space V C into the direct sum S VC DW ˚W of .i/-eigenspaces of the operator J , and this decomposition is orthogonal with respect to the Hermitian inner product h; iJ , generated by the symplectic form ! and complex structure J . The Fock space FJ D F .V C ; J / by definition coincides with the completion of the algebra S.W / of symmetric polynomials in variables z 2 W with respect to the inner product on S.W /, generated by h; iJ . The Heisenberg algebra heis.V / as a vector space coincides with V ˚ R and is provided with the Lie bracket, generated by the symplectic form !. The irreducible Heisenberg representation rJ of the Heisenberg algebra heis.V / in the Fock space FJ , which elements are identified with the holomorphic functions on S , is given in terms of the creation and annihilation operators by the formula the space W x D hz; wiJ f .w/; x aJ .z/f .w/

aJ .z/f N .w/ x D @zN f .w/ x

where z 2 W . Choosing an orthonormal basis fwn g in the space W , we introduce the operators an WD a .wn /; an WD a.w xn /; satisfying the commutation relations  ; an  D ımn Œam

(other commutators vanish). The vacuum is a vector fJ 2 FJ , annulated by all annihilation operators, i.e. an fJ D 0 for all n D 1; 2; : : : . The Fock space FJ is generated by vectors of the form .a1 /k1 : : : .an /kn fJ , which implies the irreducibility of the representation rJ . Any irreducible representation of the Heisenberg algebra, having a vacuum, is unitary equivalent to the representation r0  rJ 0 . The Shale–Berezin theorem: the Heisenberg representation rJ is unitary equivalent to the representation r0 () J 2 JHS .V /. The Fock bundle: F D

[ J 2JHS .V /

FJ ! JHS .V /

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5 Quantization of space of diffeomorphisms

is a holomorphic Hermitian Hilbert bundle, provided with a unitary projective action of the group SpHS .V /, covering its action on the base JHS .V /. The projective representation of the Hilbert–Schmidt symplectic algebra spHS .V / in the Fock space F0  FJ 0 is given by the formula   1 1 ˛ ˇ AD 7! .A/ D D˛ C Mˇ C M  N ˛ t 2 2 where

D˛ f .w/ x D h˛w; @w if .w/; x N wif .w/; Mˇ f .w/ x D hˇw; x x D h @w ; @w if .w/: x M  f .w/

The representation  intertwines with the representation r0 . The projective representation of the Hilbert–Schmidt symplectic algebra spHS .V / determines the Dirac quantization of the extended system   JHS .V / D DHS ; spHS .V /

B

B

where spHS .V / is a central extension of the algebra spHS .V / and the Dirac quantization of the system .V; A/, having the phase space given by the Sobolev space V and the algebra of observables A equal to the semi-direct sum A D heis.V / Ì spHS .V /: Using the decomposition of the Hilbert–Schmidt symplectic algebra spHS .V / D u.WC / ˚ m into the direct sum of subspaces where u.WC / is the Lie algebra of the unitary group U.WC /, one can establish a correspondence between the fprojective representations of the Lie algebra spHS .V / in the Fock space F0 g, finvariant projectively-flat connections on the Fock bundle F g and fprojective actions of the group SpHS .V / on the Fock bundle F g. Any two of these objects determine the third one. The Fock bundle over the space  F WD

[

FJ ! 

J 2

is a holomorphic Hermitian Hilbert bundle, provided with a projective action of the group DiffC .S 1 /, covering its action on the base . The projective representation of the Lie algebra VectC .S 1 / in the Fock space F0 is generated by the Virasoro operators Ln , coinciding with the images of the basis elements en of the algebra VectC .S 1 /. These operators are given by the formula 1 1 X Ln D W ai aiCn W; 2 iD1

n 2 Z:

5.2 Lecture XII. Quantization of the extended system

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This projective representation of the Lie algebraVect.S 1 / in the space F0 determines the Dirac quantization of the system .; vir/ where the Virasoro algebra vir is a central extension of the Lie algebra Vect.S 1 /.

6 Quantization of universal Teichmüller space

In this chapter we construct the quantization of the universal Teichmüller space T . First in Lecture 6.1 we define the quantization of classical systems by Connes and consider in detail (Section 6.1.2) an example of quantization of a system, having as its algebra of observables the algebra of bounded functions on the circle. Next in Lecture 6.2 we construct the Connes quantization of the system where the role of the phase space is played by the universal Teichmüller space T . The group QS.S 1 / of quasisymmetric homeomorphisms of the circle acts on this space but this action is not smooth, in particular, we can associate with it any classical algebra of observables. However, we can construct a quantum algebra of observables, associated with T . It is generated by the quantum differentials d q f , corresponding to functions f 2 QS.S 1 /. The universal Teichmüller space T , provided with this algebra of observables, should be considered as the quantum system associated with the space T .

6.1 Lecture XIII. Quantization by Connes 6.1.1 Definition. The quantization of the extended system .JHS .V /; spHS .V // and also the space of normalized diffeomorphisms .; Vect.S 1 // was based on the fact that we were able, thanks to Shale–Berezin theorem, to pull up the natural action of the group SpHS .V / on the space JHS .V / D SpHS .V /=U.WC / to the projective action of this group on the Fock bundle [ FJ ! JHS .V /: F D J 2JHS .V /

However, this method does not apply to the whole universal Teichmüller space T . Though we still have an embedding T ,! J D Sp.V /=U.WC / of the space T into the space J.V / of complex structures on V , compatible with the symplectic form !, and the Fock bundle [ FJ WD FJ ! J.V /; J 2J.V /

we cannot pull up the natural action of the group Sp.V / on J.V / to a projective action of this group on FJ , covering its action on the base J.V /. It is forbidden by the

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6 Quantization of Teichmüller space

Shale–Berezin theorem. So we have to use another approach to the quantization of T , based on ideas from the noncommutative geometry. Recall that in the Dirac approach we quantize the classical systems .M; A/, consisting of phase spaces M and algebras of observables A D AC , which are involutive Lie algebras of smooth functions on M . The quantization of such a system is given by an irreducible linear representation r of observables from A by closed linear operators, acting in the quantization space H , which transform the Poisson bracket ff; gg of observables f; g 2 A into the commutator 1i Œr.f /; r.g/ of the corresponding operators. In the Connes approach a classical system is given by the pair .M; A/ where M is again the phase space while the algebra of observables A is an associative involutive algebra consisting of smooth functions on M . The quantization of such a system by Connes is an irreducible linear representation  of observables from A by closed linear operators acting in the quantization space H , which transforms the exterior derivative operator d into the commutator with some symmetry operator S where S is a selfadjoint operator in H with square S 2 D I . In other words,  W df 7! ŒS; .f /;

f 2 A:

quantum system

classical system

We have the following table. Dirac approach

Connes approach

.M; A /

.M; A /

M – phase space

M – phase space

A – involutive Lie algebra of observables on M

A – involutive associative algebra of observables on M

irreducible representation r W A ! End H , ff; gg 7! 1i

irreducible representation W A ! End H , df 7!

Figure 2

Note that the Connes approach may be also formulated in the language of Lie algebras. Namely, consider the algebra Der.A/ of differentiations of algebra A, i.e. linear maps A ! A, satisfying the Leibniz rule. The algebra Der.A/ is a Lie algebra since the commutator of two differentiations from this algebra is again a differentiation. In terms of the algebra Der.A/ the Connes quantization is an irreducible representation of the Lie algebra Der.A/ in the Lie algebra End H , provided with the commutator as a Lie bracket.

6.1 Lecture XIII. Quantization by Connes

87

6.1.2 Comparison of the Dirac and Connes approaches. If all observables are smooth functions on M (as it was assumed before) then there is not much difference between these two approaches to the quantization. Indeed, the differential df of an observable f is symplectically dual to the Hamiltonian vector field Xf which establishes a relation between the associative algebra of observables A 3 f and the Lie algebra of Hamiltonian vector fields A 3 Xf or the dual Lie algebra of Hamiltonians f , generating vector fields Xf . The symmetry operator S is defined in this case by a polarization H D HC ˚ H (6.1) of the quantization space H , i.e. by the decomposition of H into the direct orthogonal sum of closed infinite-dimensional subspaces H˙ . The symmetry operator, corresponding to such a polarization, is equal to S D ˙I on H˙ . This operator is closely related to the complex structure operator J on H , determined by the decomposition (6.1), namely, S D iJ so that J D ˙iI on H˙ . However, if we allow the algebra of observables A to contain non-smooth functions then the Dirac definition will make no sense. In the Connes approach the differential of a non-smooth observable f 2 A will be also not defined in the classical sense, but its quantum analogue d q f WD ŒS; .f / may be correctly defined. Consider as an example the algebra A D L1 .S 1 ; C/ of bounded functions on the circle S 1 . Any function f 2 A determines the bounded multiplication operator Mf in the Hilbert space H D L2 .S 1 /, acting by the formula: Mf W h 2 H 7! f h 2 H: The symmetry operator S on H is given by the Hilbert transform Z 2 1 .S h/./ D K.; /f . /d ; f 2 H; P.V. 2 0

(6.2)

where the integral is taken in the principal value sense, i.e. "Z # Z Z 

2

K.; /f . /d

P.V. 0

WD lim

!0

2

C

K.; /f . /d : C

0

(Here and in the sequel we identify the functions f .z/ on the circle S 1 with the functions f ./ WD f .e i / on the interval Œ0; 2.) The Hilbert kernel in the formula (6.2) is given by the expression K.; / D 1 C i cot Note that for  !

it behaves like 1 C

2i 

.

 2

:

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6 Quantization of Teichmüller space

The differential of a generic observable f 2 A is not defined in the classical sense, however its quantum analogue d q f WD ŒS; Mf  is correctly defined as an operator on H (this operator is correctly defined even for functions from BMO.S 1 / introduced below). For functions f 2 V C D H01=2 .S 1 ; C/ we can assert even more. Proposition 6. A function f belongs to the Sobolev space V C if and only if its quantum differential d q f is a Hilbert–Schmidt operator on H , hence, on V C . Moreover, the Hilbert–Schmidt norm of the operator d q f coincides with the Sobolev norm of the function f . Proof. The commutator d q f WD ŒS; Mf  is an integral operator on H with the kernel equal to k.; / D K.; /.f ./  f . //: This operator is a Hilbert–Schmidt operator if and only if its kernel k.; / is square integrable on S 1 S 1 which is equivalent to the condition Z 2 Z 2 jf ./  f . /j2 (6.3)   d d < 1: sin2 2 0 0 The assertion of the proposition follows now from the Douglas formula (4.7) from Section 4.1.3. In order to see that, it is sufficient to switch in the formula (6.3) from the circle S 1 to the real line R. Then the left-hand side of the inequality (6.3) will transform into the expression Z Z   1 f .x/  f .y/ 2 dxdy D kf k21=2 ; 4 R R xy 

which implies the assertion of the proposition.

So the quantum differential d q f WD ŒS; Mf  for functions f 2 V C is an integral operator on V C , given by the formula Z 2 1 k.; /h. /d ; h 2 V C ; (6.4) .d q f /.h/./ D 2 0 where k.; / D K.; /.f ./  f . //; and K.; / is the Hilbert kernel. For  !

the kernel k.; / behaves like

f ./  f . / : 

6.1 Lecture XIII. Quantization by Connes

89

It can be shown that the limit of the operator (6.4), consisting in taking the trace of the operator (6.4) for  D , coincides with the multiplication operator h 7! f 0  h: Problem 8. Try to prove this assertion. In the considered example the quantization is reduced, essentially, to the replacement of the derivative by its finite-difference analogue. Such a quantization, given by the correspondence A 3 f 7! d q f W H ! H; ia called the “quantum calculus” by Connes by analogy with the finite-difference calculus. We give without proof several concrete examples of such a correspondence between the functions f 2 A and operators d q f on H (more detailed discussion of this correspondence may be found in [23]): 1) the differential d q f is an operator of finite rank if and only if f is a rational function (Kronecker’s theorem); 2) the differential d q f is a compact operator if and only if the function f belongs to the class VMO.S 1 /; 3) the differential d q f is a bounded operator if and only if the function f belongs to the class BMO.S 1 /. Recall for completeness the definitions of the space BMO of functions of bounded mean oscillation and the space VMO of functions of vanishing mean oscillation. It is more convenient to do it for functions f 2 L1loc .R/, defined on the real line R, rather than on the circle. Denote by Z 1 f .x/dx fI WD jI j I the average of such a function over the interval I of the real line of length jI j. If Z 1 jf .x/  fI jdx < 1 M.f / WD sup I jI j I then we shall say that the function f 2 L1loc .R/ belongs to the space BMO.R/. Introduce for ı > 0 the notation Z 1 jf .x/  fI jdx: Mı .f / WD sup jI j<ı jI j I Then f 2 BMO.R/ if and only if the function Mı .f / is bounded. We shall say that the function f 2 BMO.R/ belongs to the space VMO.R/ if Mı .f / ! 0 for ı ! 0.

90

6 Quantization of Teichmüller space

Brief content of Lecture XIII Quantization by Connes: a classical system is given in this case by the pair .M; A/ where M is the phase space, and A is an associative involutive algebra of observables. The quantization of a classical system .M; A/ is an irreducible linear representation  of observables from the algebra A by closed linear operators, acting in the quantization space H , which transforms the differential df of an observable f into the commutator d q f D ŒS; .f / of its image with a symmetry operator S, being a selfadjoint operator on H with square S 2 D I . In terms of the Lie algebras this is an irreducible representation of the Lie algebra Der.A/ of differentiations of the algebra A in the Lie algebra End H of closed linear operators in H , provided with the commutator as a Lie bracket. If all observables from the algebra A are smooth functions on M then the Connes approach is equivalent to the Dirac one. However, in the case when some of the observables are non-smooth functions the Dirac approach makes no sense. In the Connes approach the differential df of a non-smooth observable is also not defined in the classical sense but its quantum analogue d q f may be correctly defined. Example: The representation of the algebra of observables A D L1 .S 1 ; C/ in the space H D L2 .S 1 ; C/ is given by the map  W A 3 f 7! the multiplication operator Mf W H 3 h 7! f h 2 H: The symmetry operator S W H ! H coincides with the Hilbert transform and the quantum differential d q f D ŒS; Mf  is correctly defined as an operator on H . Moreover, d q f is a Hilbert–Schmidt operator () f 2 V . Quantum calculus is a dictionary, translating the properties of functions f 2 V into the properties of the corresponding operators d q f on the space H .

6.2 Lecture XIV. Quantization of the universal Teichmüller space 6.2.1 Construction of the quantization. Recall that in Section 4.1.3 we have defined a natural action of the group QS.S 1 / of quasisymmetric homeomorphisms of the circle on the Sobolev space V D H01=2 .S 1 ; R/. This action is not smooth and does not admit differentiation. We would like to associate with the space T a classical system in which T would be the phase space, and the role of the algebra of observables would be played by the Lie algebra associated with the group QS.S 1 /. However we cannot construct such a Lie algebra on the classical level because of the non-smoothness of the action of QS.S 1 /. By the same reason we cannot associate with T any classical system. However, we can associate with T a quantum system.

6.2 Lecture XIV. Quantization of the universal Teichmüller space

91

For that we define first a quantized infinitesimal action of the group QS.S 1 / on the Sobolev space V , given by the quantum differential d q , determined by the formula (6.4): QS.S 1 / 3 f 7! d q f W V ! V: Next we extend this operator to the whole Fock space F0 by defining it first on the elements of the basis PK .z/ from the formula (5.2) (Section 5.2.1) with the help of Leibniz rule, and then extending it by linearity to the whole algebra of polynomials S.WC /. The closure of the obtained operator yields an operator d q f on the Fock space F0 D S.WC /. The desired quantum Lie algebra of observables is generated by the constructed operators d q f on F0 with f 2 QS.S 1 /. We denote it by Derq .QS/ and consider as a replacement of the (non existing) classical Lie algebra of the group QS.S 1 /. Let us compare the presented method of construction of the quantum system, associated with T , with the Dirac quantization of the system .JHS .V /; spHS .V // from the Lecture 5.2:

2

JHS -case SpHS .V /-action on V

Shale– Berezin

unitary projective SpHS -action on Fock bundle infinitesimal version unitary projective representation of spHS .V / in F0

Figure 3

T-case no QS-action on Fock bundle

QS-action on V first

quantization

quantized infinitesimal action of QS.S 1 / on V

second quantization Figure 4

quantized infinitesimal action of QS.S 1 / on F0

92

6 Quantization of Teichmüller space

6.2.2 Final remarks. So, the Connes quantization of the universal Teichmüller space T consists of two stages. I. First quantization: constructing of the quantum infinitesimal action of the group QS.S 1 / on the Sobolev space V , given by QS.S 1 / 3 f 7! d q f D ŒS; Mf  W V ! V: II. Second quantization: extension of operators d q f to the Fock space F0 and construction of the quantum algebra of observables Derq .QS/, generated by operators d q f 2 End F0 with f 2 QS.S 1 /. Correspondence principle for the constructed quantization of the space T means that the restriction of the Connes quantization of T to the space of normalized diffeomorphisms   T reduces to the Dirac quantization of , constructed in Section 5.2.5. Brief content of Lecture XIV Quantization of the universal Teichmüller space: first quantization: QS.S 1 / 3 f 7! d q f D ŒS; Mf  W V ! V I second quantization: extension of the quantum differentials d q f W V ! V to the closed linear operators d q f W F0 ! F0 . The quantum algebra of observables Derq .QS/ is generated by the operators d q f W F0 ! F0 ; associated with f 2 QS.S 1 /, and may serve as a replacement of the non-existing classical Lie algebra associated with the group QS.S 1 /.

7 Instead of an afterword. Universal Teichmüller space and string theory

Our main motivation to study the universal Teichmüller space T was the relation of this space to the string theory. In a series of physical papers, devoted to this theory (cf., e.g., [24],[8]), it was noted that the space d WD C01 .S 1 ; Rd / of smooth loops in the d -dimensional vector space Rd (provided with the metric of the d -dimensional Minkowski space) may be considered as the phase space of the d -dimensional theory of closed bosonic strings. In particular, on d there is a natural symplectic form (an analogue of this loop space and its symplectic form for d D 1 was considered in Lecture 4.1, Section 4.1.1). Let us consider this interpretation of the loop space d in more detail. The configuration space of the d -dimensional bosonic theory of closed strings consists of smooth maps q W Œ0;  ! Rd such that all their derivatives vanish at boundary points. The associated phase space of this theory consists of pairs of maps .p; q/ of the same type where the map q plays the role of the “coordinate” and the map p that of the “momenta”. The symplectic form on this phase space is given by the “string analogue” of the form of type “dp ^ dq”: Z 2  !.ıp; ıq/ D ıp. / ^ ıq. /d (7.1)  0 where ıp, ıq are smooth maps Œ0;  ! Rd of the same type, as p and q, interpreted as the tangent vectors to the phase space. Consider the map, assigning to the pair .p; q/ the map x W Œ;  ! Rd , given by the formula ´ for 0    ; p. / C q 0 . / x. / D p. / C q 0 . / for     0. This map identifies the introduced phase space with the space d . It also transforms the symplectic form (7.1) on the phase space of the string theory to the symplectic form on the space d , given by the formula, analogous to (4.1) (cf. [8]): Z 2 ˛ ˝ 1 (7.2) ./; 0 . / d !.; / D 2 0 where  D .e i /, D .e i / are smooth maps Œ;  ! Rd . The symplectic form (7.2) on the space of smooth loops d may be extended to the Sobolev completion of this space, coinciding with the space of half-differentiable

94

Instead of an afterword. Universal Teichmüller space and string theory

functions Vd WD H01=2 .S 1 ; Rd /. Note that the latter space is the largest one in the scale of Sobolev spaces H0s .S 1 ; Rd / on which the form ! is correctly defined. Thus, we can say that the form ! itself “chooses” the right space to be defined on. From this point of view the choice of the space Vd as the phase space of the string theory looks more natural, compared to the conventional choice of the space d for this purpose. We set now d D 1 and denote V WD V1 D H01=2 .S 1 ; R/ to correlate with the notations and results, given in this book. By the Nag–Sullivan theorem from Section 4.1.3 there is a natural group, associated with the space V D H01=2 .S 1 ; R/, namely, the group QS.S 1 / of quasisymmetric homeomorphisms of the circle. Once again we can say that the Sobolev space V itself “chooses” the right group to act on this space. This action of the group QS.S 1 / on the space V is symplectic, i.e. it preserves the form !. If this action would be smooth then it would be natural to take for the classical system, associated with the considered string theory, the phase space V (more precisely, its d -dimensional analogue) and for its algebra of observables the Lie algebra of the group QS.S 1 /. Unfortunately, this action is not smooth, hence we cannot associate any classical system with the phase V , provided with the symplectic action of the group QS.S 1 /. However, in Section 6.2.1 it was shown that it is possible to construct a quantum algebra of observables, corresponding to this group. In terms of the twistor approach to the quantization, proposed in [27], the quantization scheme for the considered string theory is formulated as follows. We fix the phase space Vd with the symplectic action of the group QS.S 1 / on it. Consider the space of complex structures on Vd which are obtained from the original complex structure J 0 by the action of the group QS.S 1 /. This space is identified with the universal Teichmüller space T D QS.S 1 /=Möb.S 1 /. In order to quantize this theory, it is sufficient, according to [27], to construct a quantum system, associated with the space T , provided with the action of the group QS.S 1 /. This is precisely the quantum system, constructed in Section 6.2.1, with the algebra of observables, generated by the quantum differentials d q f with f 2 QS.S 1 /.

8 Problems

We have collected here the problems proposed to the listeners of the lecture course in the Educational Centre of Steklov Institute. The list contains the simple exercises, as well as “difficult” problems, but no unsolved problems. 1. Show that the dilatation of a diffeomorphism w of the complex plane, equal to Dw .z/ D

N [email protected]/j C [email protected]/j ; N [email protected]/j  [email protected]/j

is a conformal invariant, i.e. Dw .z/ D DhBwBg 1 .g.z// for any conformal maps h and g. 2. Prove the composition theorem for quasiconformal maps: the map, inverse to a K-quasiconformal map, is again K-quasiconformal and the composition of a K1 quasiconformal map with a K2 -quasiconformal one is a K1 K2 -quasiconformal map. 3. Prove the composition formula for the Beltrami differentials of quasiconformal maps:   f .z/  g .z/ @g.z/ 2 f Bg 1 .g.z// D 1  f .z/g .z/ [email protected]/j for quasiconformal maps f; g W D ! D 0 . 4. Prove the Measurable Riemann Mapping Theorem for quasiconformal maps: if x with non-trivial boundaries D and D 0 are two simply connected domains in C (i.e. the boundaries, consisting of more than one point) then for any function  2 L1 .D/ with norm kk1 < 1 there exists a quasiconformal map w W D ! D 0 with complex dilatation, equal to  almost everywhere. 5. Prove the normal convergence theorem for quasiconformal maps: if a sequence of K-quasiconformal maps wn is uniformly convergent on compacts in a domain D to a map w then w is either a constant, or a K-quasiconformal map. 6. Prove that any conformal rectangle Q.z1 ; z2 ; z3 ; z4 / may be conformally mapped onto a Euclidean rectangle ….w1 ; w2 ; w3 ; w4 / and such a transform is uniquely determined up to similarity.

96

Problems

7. Let f be a monotonically increasing homeomorphism of the real line to itself. Denote by K its maximal dilatation KD

MH.f .x1 /; f .x2 /; f .x3 /; f .x4 // MH.x1 ; x2 ; x3 ; x4 / fx1 ;x2 ;x3 ;x4 g sup

where MH.t1 ; t2 ; t3 ; t4 / denotes the conformal module of the conformal rectangle H.t1 ; t2 ; t3 ; t4 / (H is the upper halfplane). Show that the condition K < 1 is equivalent to the following condition: for some constant M > 0 the following estimate holds 1 f .x C t /  f .x/  M M f .x/  f .x  t / for all x 2 R, t > 0. 8. Show that any orientation-preserving diffeomorphism of the circle is quasisymmetric, i.e. extends to a quasiconformal diffeomorphism of the disk. x which is quasi9. Let w be a homeomorphism of the extended complex plane C conformal in the complement of some quasicircle. Then it is quasiconformal x everywhere in C. 10. Prove that the universal Teichmüller space, provided with the Teichmüller metric, is a complete linearly connected metric space. 11. Prove the existence of a function with a given Schwarzian: if ' is a given holomorphic function in a simply connected domain D then there exists a meromorphic function f in this domain with the Schwarzian, equal to ', i.e. S Œf  D '. This function f is uniquely determined by the function ' up to Möbius transformations. 12. Prove the area theorem: if f is a single-valued meromorphic function in the complement of the unit disk, having the power series decomposition of the form

then

P1 nD1

1 X bn f .z/ D z C n z nD0

njbn j2  1 and this inequality is sharp.

13. Two Möbius transformations u and v, not equal to the identity, are conjugate in the group Möb.C/ if and only if tr2 u D tr2 v. 14. Give an example of a discrete subgroup of the group Möb.C/ with the empty discontinuity domain. 15. If two Riemann surfaces X1 D Xz =G1 and X2 D Xz =G2 , uniformized by the z are biholomorphic to each other then the same universal covering surface X, groups G1 and G2 are conjugate in the automorphism group of Xz and vice versa.

97

Problems

16. Prove that the form !0 , defined on the space S 1 D C01 .S 1 ; S 1 / of smooth maps S 1 ! S 1 with zero average by the formula Z 2 1 d .e i / d; .e i / !0 .; / D 2 0 d determines a symplectic structure on S 1 , i.e. it is a closed non-degenerate 2-form. 17. Show that a linear bounded operator A W V ! V belongs to the symplectic group Sp.V / if and only if it has the block representation of the form   a b AD N with aN t a  b t bN D 1; aN t b D b t aN b aN in terms of the decomposition V C D WC ˚ W where W˙ are .i /-eigenspaces of the complex structure operator J 0 on the Sobolev space V . 18. Prove that the space J.V / D Sp.V /=U.WC / of complex structures on the Sobolev space V , compatible with the symplectic structure, may be identified with the infinite-dimensional Siegel disk D. Check that D is a convex set. 19. Show that the Hilbert–Schmidt operators T W H1 ! H2 , acting from a Hilbert space H1 to another Hilbert space H2 , form a Hilbert space. 20. Show that the Virasoro operators 1

Ln D

1X W ai aiCn W; 2 1

n 2 Z;

determine a projective representation of the Lie algebra of smooth vector fields on the circle. Concretely, ŒLm ; Ln  D .m  n/LmCn C ım;n

m3  m : 12

21. Show that the Hilbert transform determines a symmetry operator on the space L2 .S 1 ; C/. 22. Find the quasiclassical limit of the quantum differential d q f on the Sobolev space V , given by the integral operator with the kernel, equal to the finite-difference derivative of a function f 2 V . In other words, show that the trace of this operator on the diagonal for smooth functions f coincides with the operator of multiplication by the derivative f 0 . 23. Prove the Kronecker theorem: the quantum differential d q f is an operator of finite rank if and only if the function f is rational. 24. Prove that the space  of normalized diffeomorphisms of the circle is contractible.

9 Bibliographical comments

The reference list, given below, may help the interested reader to find more detailed discussion of the problems which are (by necessity) briefly presented in this book. Also, using the references, given in the text, this reader can reconstruct the details of those proofs which are given in the lectures on the level of ideas. In particular, it does not pretend to be complete. We note also that the references in the comments, given below, are given to those publications which the author found more accessible to the listeners of the course. Chapter 1. The theory of quasiconformal maps of the complex plane is considered in several books, from which we especially recommend the lectures by Ahlfors [1] where the most part of the material, given in this chapter, may be found. In particular, we follow these lectures while proving the central result of this chapter – the Existence Theorem for quasiconformal maps of the complex plane. Chapter 2. Basic notions, concerning the universal Teichmüller space, may be found in the books [16] and [18]. The problem of the existence of a Kähler metric on the universal Teichmüller space is studied in a whole series of papers. We would like to mention in this respect an interesting monograph [30] where such a metric on the universal Teichmüller space is constructed. However, the topology induced by this metric is different from the topology induced by the Teichmüller metric. (In particular, the universal Teichmüller space in the topology of [30] consists of an uncountable set of connected components.) Chapter 3. There is a large number of books, dealing with the theory of Riemann surfaces. The book [11] seems to be closest to our approach. The classical Teichmüller spaces are also studied in a whole series of books, from which we can recommend the books [16] and [18] where the exposition is close to ours. The properties of the space  of normalized diffeomorphisms of the circle are studied in the book [27] and the papers cited therein. Chapter 4. The Nag–Sullivan theorem, given in this chapter and playing a key role in the quantization of the universal Teichmüller space, is proved in the paper [20] where also the other facts, related to the embedding of the universal Teichmüller space into the infinite-dimensional Grassmannian, may be found (cf. also [22] and [27]). Chapter 5. The quantization of classical systems by Dirac is discussed in all books on geometric quantization (for example, in [33]). The Fock space and Heisenberg representation are also studied in a number of books (for example, in the books [14] and [3]). The theorem of Shale–Berezin is proved in the original paper by Shale [29] and Berezin book [3]. The projective representation of the symplectic Hilbert–Schmidt

100

Bibliographical comments

algebra is constructed in the paper [25] (cf. also [22], [27]). The representation of the Virasoro algebra is well known and presented in a whole series of books and papers (for example, in the books [14], [22]). Chapter 6. The geometric quantization by Connes is presented in his book [9] (cf. also [32]). In particular, we have borrowed from this book the example given in Section 6.1.2. The Connes quantization of the universal Teichmüller space is constructed in the paper [28] (cf. also [27]).

Bibliography

[1] L. Ahlfors, Lectures on quasiconformal mappings. Van Nostrand Mathematical Studies 10, Van Nostrand, Princeton, NJ, 1966. [2] L. Ahlfors, Conformal invariants: topics in geometric function theory. McGraw-Hill Series in Higher Mathematics, McGraw Hill, New York 1973. [3] F. A. Berezin, Method of second quantization. Pure and Applied Physics 24, Academic Press, New York 1966. [4] L. Bers, Automorphic forms and general Teichmüller spaces. In Proceedings of the Conference on Complex Analysis, Minneapolis, 1964 (eds. A. Aeppli, E. Calabi, H. Röhrl), Springer-Verlag, Berlin 1965, 109–113. [5] L. Bers, Universal Teichmüller space. In Analytic methods in mathematical physics, Symposium held at Indiana University, Bloomington, Ind., 1968 (eds. R. P. Gilbert, R. G. Newton), Gordon and Breach, New York 1970, 65–83. [6] L. Bers, Uniformization, moduli, and Kleinian groups. Bull. London Math. Soc. 4 (1972), 257–300. [7] R. Bowen, Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 259–273. [8] M. J. Bowick and S. G. Rajeev, The holomorphic geometry of closed bosonic string theory and Diff S 1 =S 1 . Nuclear Phys. B 293 (1987), no. 2, 348–384. [9] A. Connes, Geometrie non commutative. InterEditions, Paris 1990. [10] A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle. Acta Math. 157 (1986), 23-48. [11] H. M. Farkas and I. Kra, Riemann surfaces. Second edition. Grad. Texts in Math. 71, Springer-Verlag, New York 1992. [12] F. P. Gardiner and D. Sullivan, Symmetric and quasisymmetric structures on closed curve. Amer. J. Math. 114 (1992), 688–736. [13] R. Goodman and N. R. Wallach, Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math. 347 (1984), 69–133; Erratum ibid. 352 (1984), 220. [14] V. G. Kac and A. K. Raina, Highest weight representations of infinite dimensional Lie algebras. Advanced Series in Mathematical Physics 2, World Scientific Publishing, Teaneck, NJ, 1987. [15] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Volume I, Interscience Publishers, New York 1963. [16] O. Lehto, Univalent functions and Teichmüller spaces. Grad. Texts in Math. 109, SpringerVerlag, New York 1987. [17] O. Lehto and K. Virtanen, Quasiconformal mappings in the plane. Grundlehren Math. Wiss. 126, Springer-Verlag, Berlin 1973.

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[18] S. Nag, The complex analytic theory of Teichmüller spaces. Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley Interscience, New York 1988. [19] S. Nag, A period mapping in universal Teichmüller space. Bull. Amer. Math. Soc. 26 (1992), 280–287. [20] S. Nag and D.Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the H 1=2 space on the circle. Osaka J. Math. 32 (1995), 1–34. [21] S. Nag and A. Verjovsky, Diff.S 1 / and the Teichmüller spaces. Comm. Math. Phys. 130 (1990), 123–138. [22] A. Pressley and G. Segal, Loop groups. Oxford Math. Monogr., Clarendon Press, Oxford 1986. [23] S. Power, Hankel operators on Hilbert space. Research Notes in Math. 64, Pitman, Boston, MA, 1982. [24] J. Scherk, An introduction to the theory of dual models and strings. Rev. Mod. Phys. 47 (1975), 123–164. [25] G. Segal, Unitary representations of some infinite dimensional groups. Comm. Math. Phys. 80 (1981), 301–342. [26] A. G. Sergeev, Lectures on universal Teichmüller space. Lecture Course of SEC, Steklov Mathematical Institute, Moscow 2013 (in Russian). [27] A. Sergeev, Kähler geometry of loop spaces. World Scientific Publishing, Singapore 2011. [28] A. G. Sergeev, The group of quasisymmetric homeomorphisms of the circle and quantization of the universal Teichmüller space. SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), 015, 1–20. [29] D. Shale, Linear symmetries of free boson field. Trans. Amer. Math. Soc. 103 (1962), 149–167. [30] L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space. Mem. Amer. Math. Soc. 183 (2006), no. 861. [31] H. Triebel, Theory of function spaces. Mathematik und ihre Anwendungen in Physik und Technik 38, Akademische Verlagsgesellschaft, Leipzig 1982. [32] J. M. Gracia-Bondia, J. C. Várilly, and H. Figueroa, Elements of noncommutative geometry. Birkhäuser Adv. Texts, Birkhäuser, Boston, MA, 2001. [33] N. M. J. Woodhouse, Geometric quantization. 2nd ed., Oxford Math. Monogr., Clarendon Press, Oxford 1992.

Index

absolutely continuous function on rectangles (RAC), 3 Ahlfors map, 32 algebra of differentiations, 86 algebra of observables, 70 annihilation operator, 74 Area Theorem, 27 Beltrami differential, 4 Beltrami equation, 4 Bers embedding, 28 Beurling–Ahlfors theorem, 14 Calderon–Zygmund type integral operator, 7 Cauchy–Riemann equation, 4 classical system, 69 classical Teichmüller space, 45 complex dilatation, 4 Composition Theorem, 5 conformal map, 1 conformal modulus, 12, 13 conformal rectangle, 12 conformal type of a Riemann surface, 41 correspondence principle, 92 creation operator, 74 cross ratio, 15 differential of type .m; n/, 4 Dirichlet fundamental domain, 40 Dirichlet space, 57 discontinuity domain, 38 double ratio, 15 Douglas formula, 60 elementary Kleinian group, 39 elliptic fractional-linear transformation, 38 energy of harmonic function, 57

Existence Theorem, 6, 11 Fock bundle, 76 Fock space, 73 Fredholm operator, 66 Fuchsian group, 39 Fuchsian group of the first kind, 39 Fuchsian group of the second kind, 39 fundamental domain, 39 G-invariant quasiconformal homeomorphism, 44 G-invariant quasisymmetric homeomorphism, 45 Heisenberg algebra, 70, 73 Heisenberg representation, 73 Hilbert kernel, 87 Hilbert transform, 50, 87 Hilbert–Schmidt complex structure, 67 Hilbert–Schmidt Grassmannian, 66 Hilbert–Schmidt operator, 66 Hilbert–Schmidt Siegel disk, 67 Hilbert–Schmidt symplectic group, 67 hyperbolic fractional-linear transformation, 38 Kleinian group, 39 K-quasiconformal diffeomorphism, 2 K-quasiconformal homeomorphism, 3 limit set of Kleinian group, 39 loxodromic fractional-linear transformation, 38 Lp -solution of Beltrami equation, 4 maximal dilatation, 13, 14 M -quasisymmetric homeomorphism, 14 non-elementary Kleinian group, 39

104 normal solution of Beltrami equation, 6 normalized quasisymmetric diffeomorphisms, 20 normalized quasisymmetric homeomorphisms, 20 parabolic fractional-linear transformation, 38 phase space, 69 polarization, 87 properly discontinuous action, 38 quantization by Connes, 85 quantization by Dirac, 70 quantization space, 70 quantized infinitesimal action, 91 quantum algebra of observables, 91 quasicircle, 21 quasiconformal diffeomorphism, 1 quasiconformal homeomorphism, 3 quasidisk, 21 quasisymmetric homeomorphism of the circle, 15 quasisymmetric vector fields, 33 regular point of T , 47

Index

Schottky double, 42 Schwarz formula, 50 Schwarzian derivative, 25 Segal theorem, 77 Shale–Berezin theorem, 75 Siegel disk, 65 Sobolev space of half-differentiable functions, 55 symmetry operator, 86 symplectic group of a Hilbert space, 64 Teichmüller distance, 24 Teichmüller lemma, 31 Uniqueness Theorem, 6 universal Teichmüller space, 19 vacuum, 74 Virasoro algebra, 80 Virasoro operators, 79 Virasoro representation, 79 Weil–Petersson metric, 47 Zigmund space, 33