Classical and quantum Teichmüller spaces

In his famous paper [93] Georg Friedrich Bernhard Riemann (1826–1866) counted the number of parameters for the isomorphism classes of algebraic equations in two complex variables or, in other words, for the classes of compact Riemann surfaces up to biholomorphic equivalence. This brought him to the definition of what is now called the coarse moduli space: the global parameter space of complex structures on a 2-dimensional surface. About 80 years later, Teichmüller understood that classification problems are easier to treat if the objects in question are endowed with some additional structure, a framing. His idea was to start with the classification of the 'marked objects' and then describe the equivalence relations arising when the framing is 'forgotten'.

Here we review briefly the life of Teichmüller and other 'actors' in the theory considered in this survey.

Paul Julius Oswald Teichmüller (1913–1943) enrolled in Göttingen University to study mathematics in the summer of 1931, and immediately distinguished himself as an extraordinary student. Helmut Hasse (1898–1979), who took over Hermann Weyl's chair after his death in 1934, became Teichmüller's scientific advisor. Although he chose Hasse to guide him, Teichmüller's interests in mathematics were far away from the lines developed by Hasse.

Unfortunately, Teichmüller's interests were not confined to mathematics. In Göttingen he fell under the influence of Nazi propaganda and joined the Nazi Party. After also joining the SA (Sturmabteilung), he became a leader of students sympathizing with the Nazis. It was he who led the students' boycott on 2 November 1933 of Edmund Landau's lectures, in the spirit of Hitler's notorious Berufsbeamtengesetz.

In 1935 Teichmüller defended his thesis for the degree of Doktor der Naturwissenschaften and in 1936, after moving to Berlin, he started working on his Habilitationsschrift under the auspices of Ludwig Bieberbach (1886–1982). He defended this thesis [106] in March 1938. (We note in parentheses that Bieberbach did not like Teichmüller's early algebraic papers written under Hasse's influence.)

From 1935 through 1943, that is, in less than 10 years, he published 34 papers which in many respects anticipated the impressive development of the theory of Riemann surfaces over the next 70 years. Teichmüller's interests encompassed a wide range of branches of mathematics, from mathematical logic, through algebra, number theory, and function theory, to topology and differential geometry. Starting from his D.Sc. thesis [106] and the subsequent paper [107], and then in the famous paper [108] and its supplement [109], he laid the foundations for what is now called the theory of Teichmüller spaces. One of his last papers developing this theory, [110], was published after his death.

Among these 34 papers, 21 were published in Deutsche Mathematik, a mathematical journal founded by Bieberbach and Theodor Vahlen in 1936 "to promote German mathematics as a counterbalance to Jewish mathematics". Since most of Teichmüller's papers were published in this journal, which was discontinued after World War II, they remained almost inaccessible until 1986, when Lars Ahlfors and Frederick Gehring published a volume of selected papers of Teichmüller [111].

As a member of the Nazi Party, Teichmüller was actively engaged in battles of the war, from the 1940 Nazi occupation of Norway onwards. In the early 1940s, under Bieberbach's influence, he turned to cryptography and even gave lectures on the topic in Berlin in 1942. However, after the 1942 defeat of Nazi troops in Stalingrad, he returned to the Wehrmacht and was sent to the Eastern Front, where he was killed on 11 September 1943, near Poltava, in the wake of the last battle of Kharkov.

More details about Teichmüller's work and life can be found in the publications [4], [81], [95], and also in [23].

Passing to publications concerned with Teichmüller theory, we start with the aforementioned selection [111] of Teichmüller's papers. Besides this we point out the Handbook of Teichmüller theory [84], the first two volumes of which give a deep and comprehensive survey of Teichmüller theory (with more than 1500 pages). There are several monographs devoted partially or wholly to various aspects of this theory, for example, [3], [9], [30], [32], [33], [43], [45], [58], [62], [66], [71], [97], [113], [118]. Surprisingly, not so many surveys present the current progress in this theory. In this connection we note [85], which treats mostly the universal Teichmüller space introduced by Bers [17] in 1968.

As is well known, Riemann [93] already observed that the conformal equivalence classes of genus-$g$ surfaces depend on $6g-6$ real parameters. In 1939 Teichmüller [108] continued (mostly on a heuristic level) the investigation of these equivalence classes on the basis of an entirely new idea. Namely, let $S_1$ and $S_2$ be homeomorphic Riemann surfaces of genus $g$ and let $f$ be a homeomorphism from $S_1$ onto $S_2$. Assume that $S_1$ and $S_2$ are not conformally equivalent. Then there must exist quasiconformal maps from $S_1$ onto $S_2$ which are homotopic to $f$. In the class of these maps there exists a unique map $f_0$ minimizing the real dilatation $K(f)$ (which characterizes the maximum deviation of $f$ from a conformal map). Now we identify $S_1$ and $S_2$ with points in some metric space endowed with the metric $(1/2)\log K(f_0)$. This metric space is locally homeomorphic to the ($6g-6$)-dimensional Euclidean space. In [108] Teichmüller managed to prove the uniqueness and then in 1943 the existence of such a map $f_0$ [109].

It should be pointed out that Teichmüller's original proofs were mostly heuristic and contained serious gaps. Ahlfors and Bers expended much effort to convert these proofs into rigorous arguments. First in 1953 Ahlfors [7] presented his own proof of the existence theorem and then Bers [16] was able to transform Teichmüller's original heuristic ideas into a framework of rigorous mathematics.¹

The influence of these papers by Ahlfors and Bers on the mathematical community was so great that a series of colloquia devoted to their mathematical heritage and called the Bers Colloquia were organized at the Graduate Center of the City University of New York, starting in 1955 and held every three years. After Ahlfors' death the name was changed to the Ahlfors–Bers Colloquia. (The first colloquium bearing this name was held in 1998 at Stony Brook.)

Lars Valerian Ahlfors was born in Helsingfors (now Helsinki), Finland, in 1907, in a Swedish-speaking family. His father was a professor of engineering at the Institute of Technology, and his mother died tragically at his birth. In 1924–1928 Ahlfors studied mathematics at the University of Helsinki, where Lindelöf and Rolf Nevanlinna were his teachers. In 1928–1929 Nevanlinna was offered Hermann Weyl's chair at the Zürich Eidgenössische Technische Hochschule and he proposed that Ahlfors go with him. In 1930 Ahlfors next moved to Paris together with Nevanlinna. In the same year 1930 he submitted his Ph.D. thesis and set out for a European trip. In 1935 he met Carathéodory, who dispatched him to Harvard, for a three-year teaching stint. However, he became homesick and returned to Finland in 1938 upon being offered a mathematics chair at Helsinki University. At the beginning of the war he and his family moved to Sweden.

While in Sweden Ahlfors met Arne Beurling, who was of great help to him on many occasions. Subsequently they developed a friendship which lasted many years, until Beurling's death in 1986. However, there was a long break in this friendship, which occurred in the following circumstances. Beurling was a very scrupulous researcher and never published results without being certain of their absolute correctness. For this reason, when Ahlfors mentioned their recent joint work in one of his lectures, Beurling was upset, for he did not consider the work completed. Although Ahlfors' actual intention was to underscore Beurling's role in the proof, the two scholars became estranged and only renewed their friendship two years before Beurling's death.

In 1944 Ahlfors was offered a chair in Zürich. However, it proved impossible for him to travel there that year, and only in 1945 did he manage to get to Zürich via Prestwick, Glasgow, London, and Paris. He did not like his stay in Zürich because, in his words, the post-war era was not a good time for a stranger to take root in Switzerland. Therefore, he gladly accepted an offer from Harvard, where he worked until retiring in 1977. Ahlfors died of pneumonia in Pittsfield, Massachusetts, on 11 October 1996, at the age of 89. His books Complex analysis (1953), Riemann surfaces (jointly with L. Sario) (1960), Lectures on quasiconformal maps (1966), and Conformal invariants (1973), as well as his papers, which number more than a hundred, still retain their value.

Ahlfors was awarded several honours for his outstanding contributions to mathematics. Among these was the very first Fields Medal in 1936 (just four years after John Fields' death) for his work in the theory of Riemann surfaces and quasiconformal maps. Another great honour was the Wolf Prize awarded to him in 1981. He was the author of fundamental results on meromorphic curves, value distribution, Riemann surfaces, conformal geometry, quasiconformal maps, Kleinian groups, and in other areas of mathematics.

Lipman Bers, known to his friends as Lipa, was born in a Jewish teacher's family in Riga on 23 May 1914. At the time of World War I and the Russian Revolution, Latvia was affected by civil war, and Bers' family moved to St.-Petersburg (called Petrograd at the time), but returned to Latvia in 1918, when it became independent. On 15 May 1934, Ulmanis led a bloodless coup and established a nationalist authoritarian regime in Latvia which lasted until 1940. Since Bers was a person of social-democratic views and an ardent civil rights advocate, an arrest warrant for him was issued, and he had to flee to Prague. His girlfriend Mary Kagan followed him, and they married there on 15 May 1938.

In the same year 1938, Bers got his Ph.D. from Charles University in Prague, under the supervision of Karel Löwner (known in the history of mathematics as Charles Loewner, the name he took after emigrating to the USA). However, he was not satisfied with his collaboration with Loewner since he did not feel Loewner's support. Bers recollected that only in retrospect did he understand Loewner's teaching method. He gave to each of his students only the amount of help needed $\dots$. Obviously, Loewner did not think that Lipa was in any great need of support (see [82]). Since both Loewner and Bers were Jewish, it was dangerous for them to remain in Prague after the Nazi occupation of Czechoslovakia in 1938. In the end Loewner was imprisoned. Bers was sought by Latvian authorities, so he preferred to move to Paris, where his daughter Ruth was born. As Nazi troops were approaching Paris, Bers applied to the American embassy for a visa, but had no time to get it and had to flee to the unoccupied part of France.

Only in 1940 did Bers and his family arrive in the United States, where they reunited with his mother, who was already in New York. It can hardly be said that scientists from Nazi Germany and occupied countries were received with open arms in the US. There were not enough vacancies at universities, and even the brightest of the newly arrived researchers could not find jobs conforming to their talents. For this reason Lipa remained unemployed until 1942, when he was offered a research instructor position at Brown University. In 1945–1949 Bers and Loewner were invited to Syracuse University. At that time William T. Martin from MIT became the head of the mathematics department of Syracuse University. He was to reorganize the department, which hitherto had been scattered among the Schools of Arts and Sciences, Business, and Engineering. This was a period in which the department flourished, because Martin invited a whole group of brilliant mathematicians such as Loewner, Bers, and Abe Gelbart from Brown University.

At that time the main subject of Bers' investigations was the problem of removal of singularities of solutions of non-linear elliptic equations. He presented these results at the International Congress of Mathematicians in 1960, and his paper [15] was published in Annals of Mathematics in 1951. Bers spent the years 1949–1951 at the Princeton Institute for Advanced Study, where he started his 10-year mathematical odyssey which took him ultimately to the theory of pseudoanalytic functions, quasiconformal maps, Teichmüller theory, and Kleinian groups. In 1951 he moved to New York University's Courant Institute and in 1964 to Columbia University, where he remained until his retirement in 1984. He died on 29 October 1993, after a long struggle with Parkinson's disease.

He was the author of fundamental results in several areas of mathematics, including partial differential equations, quasiconformal maps, Teichmüller theory, and Kleinian groups (see [5] for details). During his life he held a series of prominent mathematical positions, in particular, the vice-presidency of the American Mathematical Society from 1963 to 1965 and its presidency from 1975 to 1977. He was awarded the Steel Prize in 1975, and in 1967 the London Mathematical Society invited him to give the first Hardy Lectures. Bers was very involved in political and social activities, always struggling for political and civil rights.

The history of Riemann surfaces apparently began in 1842, when Karl Weierstrass stated the analytic continuation principle [119]. In his Doktor der Naturwissenschaften thesis [92] Riemann, who was a student of Gauss, first introduced topological methods in function theory, thereby ushering in the theory of the surfaces which now bear his name. In contrast to Weierstrass' approach, Riemann's definition was based on the concept of a covering. (An interesting attempt to make Riemann's concept clear to a wide audience was made by Carl Neumann in 1865 [77].)

The theory of Riemann surfaces was further developed by Felix Klein, who in 1882 introduced into mathematics the notion of an atlas [55]. In his 1881 paper [86] and later publications, Henri Poincaré laid the grounds for uniformization theory and the group approach in the theory of Riemann surfaces. Adolf Hurwitz established links between this theory and the theory of algebraic curves [44]. Hermann Weyl's 1913 book Die Idee der Riemannschen Fläche [121] had a very strong impact on the further development of the theory. It contained a systematic introduction to such concepts as a covering surface, the deck transformation group, a simply connected domain, and the genus.

Many well-known authors subsequently contributed to the theory. Here we must mention Koebe, Radó, Stein, Stoïlov, Springer, and others. In particular, in 1925 Tibor Radó [91] first proposed the definition of a Riemann surface as a topological surface with a complex structure and proved that such a surface can be triangulated. In the 25 years that followed, the theory of abstract Riemann surfaces experienced unprecedented growth. A notable contribution to it was made by Rolf Herman Nevanlinna (1895–1980), one of the best known Finnish mathematicians. In his 1953 monograph Uniformisierung [78], based on lecture courses he gave in Zürich and Helsinki, Nevanlinna presented the contemporary concept of abstract Riemann surfaces in concentrated form, focusing mostly on the case of non-compact ('open') Riemann surfaces.

A Riemann surface $S$ is a connected Hausdorff topological space $M$ endowed with a given open covering $\lbrace U_j\rbrace$ and a system of homeomorphisms $\lbrace g_j\rbrace$, $g_j:U_j\rightarrow \mathbb {C}$, such that the composite maps $g_j\circ g_k^{-1}$ are conformal on the non-empty intersections $U_j\cap U_k$. One says that the set of pairs (charts) $\lbrace U_j,g_j\rbrace$ defines a conformal structure on $S$. Two structures $\lbrace U^1_j,g^1_j\rbrace$ and $\lbrace U^2_j,g^2_j\rbrace$ are said to be equivalent if their union also defines a conformal structure on $S$. Thus, two Riemann surfaces with the same topological base space and equivalent conformal structures define the same Riemann surface. The space $S$ can also be viewed as a connected component of a 1-dimensional complex topological manifold (which is therefore an oriented manifold).

The fundamental group of a Riemann surface plays a central part in the theory. Fix a point $p\in S$ and look at the class $[c]$ of homotopic closed curves on $S$ (that is, continuous maps from the interval $[0,1]$ to $S$) with initial point and endpoint at $p$ and with given orientation. On the curves in $[c]$ we have the operation of taking the composition $c_1c_2$ of two loops $c_1$ and $c_2$, which is defined by going along the oriented curves one after the other, and the operation of taking the inverse $c^{-1}$, defined by going along the curve $c$ in the reverse direction. These operations extend to corresponding operations on homotopy classes: $[c_1][c_2]=[c_1c_2]$ and $[c]^{-1}=[c^{-1}]$. The set of these classes with operations on it is called the fundamental group $\pi _1(S,p)$ of $S$ with base point $p$. A curve $\gamma$ joining a point $p$ to another point $p_1$ defines an isomorphism $[\gamma c\gamma ^{-1}]$ of the fundamental groups $\pi _1(S,p)$ and $\pi _1(S,p_1)$. With this isomorphism in mind, we can speak about the single fundamental group $\pi _1(S)$ of the surface $S$.

For an arbitrary subgroup $F$ of the fundamental group $\pi _1(S)$ of a Riemann surface $S$ we can construct another Riemann surface $S_F$ which is an unbounded covering space of $S$ such that $\pi _1(S_F)=F$. If $F$ is the trivial subgroup $\lbrace 1\rbrace$, then the covering surface $\widetilde{S}\equiv S_1$ is called the universal covering of $S$. The surface $\widetilde{S}$ is simply connected, and by the Poincaré–Koebe uniformization theorem there is a conformal homeomorphism $h$ taking $S$ onto one of the canonical domains $\mathbb {D}=\lbrace z\in \mathbb {C}:|z|<1\rbrace$, $\mathbb {C}$, or $\widehat{\mathbb {C}}$. Accordingly, the surface $S$ is said to be hyperbolic, parabolic, or elliptic. In particular, a simply connected hyperbolic domain in $\widehat{\mathbb {C}}$ must have more than two boundary points.

A deck transformation of the Riemann surface $\widetilde{S}$ induces a conformal automorphism of the canonical domain. Let $G$ be the automorphism group of the canonical domain corresponding to the deck transformation group of $\widetilde{S}$. The composite $J\circ h^{-1}$ with the projection $J:\widetilde{S}\rightarrow S$ is an automorphic analytic function with respect to $G$. We can identify a universal covering space with its $h$-image. Then the Riemann surface $S$ can be identified with the quotient $S=D/G$, where $D$ is a canonical domain up to conformal equivalence. In this case we say that the group $G$ uniformizes the surface $S$.

An elliptic Riemann surface is conformally equivalent to the Riemann sphere $\widehat{\mathbb {C}}$.

Any parabolic Riemann surface is conformally equivalent to the plane $\mathbb {C}$, the punctured plane $\mathbb {C}\setminus \lbrace 0\rbrace$, or a torus. In this case the group $G$ is trivial, or consists of the transformations of the form $z^{\prime }=z+n\omega$, or consists of the transformations of the form $z^{\prime }=z+n\omega _1+m\omega _2$, where $\omega$, $\omega _1$, and $\omega _2$ are some complex numbers such that $\operatorname{Im}(\omega _2/\omega _1)\ne 0$, and $n$ and $m$ are integers. All the other Riemann surfaces have hyperbolic type, and the corresponding group $G$ is a discrete subgroup of the Möbius group of conformal automorphisms of the disk $\mathbb {D}$. Recall that these automorphisms are given by the formula

$$\begin{equation*} z\mapsto e^{i\theta }\frac{z-a}{1-z\bar{a}},\qquad a\in \mathbb {D},\quad 0\leqslant \theta <2\pi . \end{equation*}$$

In this case $G$ is called a Fuchsian group. The fundamental domain $D$ of a Fuchsian group $G$ is a subregion of $\mathbb {D}$ consisting of points that cannot be taken one to another by an action of $G$. Geometrically, this domain is a hyperbolic polygon in $\mathbb {D}$ bounded by horocycle arcs and pieces of the boundary $\partial \mathbb {D}$. If this polygon has finitely many sides, then $G$ is finitely generated. A Riemann surface $S$ with a finitely generated uniformizing group $G$ is otherwise called a finite Riemann surface. A compact Riemann surface is finite and is topologically equivalent to a sphere with finitely many handles, and the number $g$ of handles is called the genus of the surface. By its definition, the genus is a topological invariant of the surface. If it has genus $g$, then we can choose generators $A_1,B_1,\dots ,A_g,B_g$ of $G$ which have hyperbolic fixed points on the boundary $\partial \mathbb {D}$ and correspond to the sides of the polygon $D$. In addition, if a surface has $n$ punctures, then we can add generators $C_1,\dots ,C_{n}$ corresponding to $n$ parabolic fixed points (cusps on $\partial D$). These generators satisfy the unique relation

$$\begin{equation*} C_{n}\circ \dots \circ C_1\circ B_g^{-1} \circ A_g^{-1}\circ B_g\circ A_g \circ \dots \circ B_1^{-1}\circ A_1^{-1} \circ B_1\circ A_1=\operatorname{id}. \end{equation*}$$

If $S$ also has $l$ hyperbolic boundary components, then it is called a surface of finite type $(g,n,l)$.

Thus, a Riemann surface $S$ has finite type $(g,n)$ if $S=\widehat{S}\setminus F$, where $\widehat{S}$ is a compact Riemann surface of genus $g$ and $F$ is an $n$-point set. Small neighbourhoods of the boundary points of $S$ are punctured disks. By a Riemann surface $S$ of finite type $(g,n,m)$ we mean a surface of the form $S=S^{\prime }\setminus E$, where $S^{\prime }$ is a Riemann surface of finite type $(g,n)$ and $E$ is a system of $m$ disks embedded in $S^{\prime }$.

Quasiconformal maps were introduced in 1928 by Herbert Grötzsch (1902–1993) [38], and then defined again in 1935 by Lars Ahlfors [6] as homeomorphisms of plane domains which to first order take small circles to small ellipses with uniformly bounded eccentricities. We remark that Ahlfors was awarded the Fields Medal just for his investigations in the theory of quasiconformal maps. Quasiconformal maps in dimension 3 were introduced by Mikhail Alekseevich Lavrent'ev (1900–1980) in his 1938 papers [64] and [65], where he studied solutions of elliptic equations.

We start with the definition of quasiconformal maps of plane domains. Let $D$ be a domain in $\widehat{\mathbb {C}}$ (perhaps coinciding with $\widehat{\mathbb {C}}$) and $w=f(z)$ a homeomorphism of $D$ onto another domain $D^{\prime }\subseteq \widehat{\mathbb {C}}$. Let

$$\begin{equation*} f_{\bar{z}} :=\frac{\partial f}{\partial \bar{z}} =\frac{1}{2}\biggl (\frac{\partial f}{\partial x} +i\frac{\partial f}{\partial y}\biggr ),\quad f_z:=\frac{\partial f}{\partial z} =\frac{1}{2}\biggl (\frac{\partial f}{\partial x} -i\frac{\partial f}{\partial y}\biggr ),\qquad z=x+iy, \end{equation*}$$

be distributional derivatives of $f$ that are locally square-integrable functions on $D$. Then $f$ is called a quasiconformal diffeomorphism of $D$ if the complex function $\mu _f(z)=f_{\bar{z}}/f_z$ satisfies an inequality $|\mu _f(z)|\leqslant k<1$ almost everywhere in $D$. Moreover, $f$ is called a $K$-quasiconformal diffeomorphism, with $K$ defined by $K=(1+k)/(1-k)$. The function $\mu _f(z)$ is called the complex characteristic or complex dilatation of $f$. Note that a quasiconformal map is conformal if and only if $K=1$ (or $k=0$).

The following notation will be convenient in what follows. In the case of the Riemann sphere we let $\widehat{\mathbb {C}}_K$ denote the class of $K$-quasiconformal automorphisms $f:\widehat{\mathbb {C}}\rightarrow \widehat{\mathbb {C}}$, normalized by the conditions $f(0)=0$, $f(1)=1$, and $f(\infty )=\infty$. In the case of the unit disk we let $\mathbb {D}_K$ denote the class of $K$-quasiconformal automorphisms $f:\mathbb {D}\rightarrow \mathbb {D}$ with the normalization $f(\pm 1)=\pm 1$, $f(i)=i$.

A quasiconformal map is a homeomorphic generalized solution of Beltrami's equation

$$\begin{equation} w_{\bar{z}}=\mu _f(z)w_z. \end{equation} \tag{ 1 }$$

This solution is uniquely defined up to conformal homeomorphisms. By adding some condition of conformal normalization (for example, fixing three boundary points of a simply connected domain) we can attain uniqueness of the solution of (1).

Turning to the geometric definition of quasiconformal maps, we use the notion of the modulus of a family of curves. In terms of this notion an orientation-preserving homeomorphism $f$ of a domain $D\subset \widehat{\mathbb {C}}$ onto a domain $D^{\prime }\subset \widehat{\mathbb {C}}$ is said to be $K$-quasiconformal if for any doubly connected hyperbolic domain $R\subset D$ the ratio $M(f(R))/M(R)$ is bounded and

$$\begin{equation} \frac{M(R)}{K}\leqslant M(f(R))\leqslant KM(R), \end{equation} \tag{ 2 }$$

where $M(R)$ is the conformal modulus of a doubly connected domain. The inequality (2) is called the quasi-invariance of the modulus.

Now we look more closely at the connection between a quasiconformal map $f$ and $\mu _f(z)$. The dilatation $\mu _f(z)$ is a bounded measurable function of $z$ with norm $\Vert \mu _f\Vert _\infty <1$. If $\mu _f(z)$ is an $n$-fold differentiable function of $z$ with $n\geqslant 1$, and its $n$th derivative is Hölder continuous with exponent $\alpha \in (0,1)$, then a quasiconformal solution $f^\mu \in \mathbb {D}_K$ of (1) is $(n+1)$-fold differentiable, and its $(n+1)$st derivative is Hölder continuous with the same exponent [13]. One might think that $f^\mu$ is continuously differentiable for any continuous function $\mu$, but this is not true, as we see from the simple example of the map $f(z)=z(1-\log |z|)$, $f(0)=0$, $z\in \mathbb {D}$, which is not even Lipschitz continuous at $z=0$. Belinskii [13] showed that if $\mu$ is continuous, then the solution $f$ is Hölder continuous with any exponent $0<\alpha <1$.

Assume that the dilatation $\mu _f(z,t)$ of a quasiconformal homeomorphism $f$ of the unit disk $\mathbb {D}$ depends on a real or complex parameter $t$. If $\mu (\,\cdot \,,t)$ is a continuous, differentiable, or holomorphic function of $t$, then so is the solution $f^\mu$.

We now turn to quasiconformal maps between Riemann surfaces. Let $S_0=\mathbb {D}/G_0$ and let $\omega$ be a quasiconformal automorphism of $\mathbb {D}$. Let $m_\omega$ be the dilatation of $\omega$. Then

$$\begin{equation*} m_\omega (\gamma (z)) \frac{\overline{\gamma ^{\prime }(z)}}{\gamma ^{\prime }(z)} =m_\omega (z)\qquad \text{for } \gamma \in G_0. \end{equation*}$$

We define the Fuchsian group $G=\omega \circ G_0\circ \omega ^{-1}$ and let $S$ denote the Riemann surface $S=\mathbb {D}/G$. Then we have the commutative diagram

in which $f=J\circ \omega \circ J_0^{-1}$ is called a quasiconformal homeomorphism of $S_0$ onto $S$, and $J$ and $J_0$ are automorphic projections. The dilatation of $f$ (or the Beltrami differential $\mu _f$) in terms of the local parameter $\zeta$ is

$$\begin{equation*} \mu _f(\zeta )=m_\omega (J_0^{-1}(\zeta )) \frac{\overline{(J_0^{-1}(\zeta ))^{\prime }}}{(J_0^{-1}(\zeta ))^{\prime }}. \end{equation*}$$

The Fuchsian groups $G_0$ and $G$ are the automorphism groups of the functions $J_0$ and $J$, respectively, and $f$ induces a group isomorphism $\chi _f:G_0\rightarrow G$. The dilatations $m_\omega$ corresponding to various $\omega$ fill the whole unit ball $D(G_0)$ in the space $B(G_0)$ of $G_0$-invariant Beltrami differentials with finite norm $\Vert \,\cdot \,\Vert _\infty$.

A holomorphic (meromorphic) quadratic differential is a holomorphic (meromorphic, respectively) symmetric 2-form on a Riemann surface $S$. It assigns to each point $p$ on $S$ a $(0,2)$-tensor $\varphi =\varphi (z)\,dz^2$, where $z$ is a local coordinate at $p$. If $\varphi ^*$ and $\varphi$ are representations of the same differential in terms of local parameters $\zeta ^*$ and $\zeta$, then

$$\begin{equation*} \varphi ^*(\zeta ^*) =\varphi (\zeta ) \biggl (\frac{d\zeta }{d\zeta ^*}\biggr )^2, \end{equation*}$$

where the change of variables $\zeta =\zeta (\zeta ^*)$ is defined in the intersection of the corresponding coordinate neighbourhoods.

Let $H^{2,0}(S_0)$ be the space of holomorphic quadratic differentials with finite $L^1$-norm on a surface $S_0$ of type $(g,n)$. In particular, such a differential can at worst have simple poles on the compactification $\widehat{S}_0$ of $S_0$. Hence by the Riemann–Roch theorem,

$$\begin{equation*} \dim _{\mathbb {C}}H^{2,0}(S_0) =\left\lbrace \begin{array}{@{}l@{\quad }l@{}}n-3, &g=0, \\ 1, &g=1,\quad n=0, \\ n-1, &g=1,\quad n>0, \\ 3g-3+n, &g\geqslant 2. \end{array}\right. \end{equation*}$$

Let us introduce the conformally invariant metric generated by the quadratic differential $\varphi$. The length element in this metric is $|dw|=\sqrt{|\varphi (\zeta )|}\,|d\zeta |$. Calculating the quadratic differential $\varphi$ on a given curve $\gamma =\gamma (t)$, we obtain a function of $t$ which can be naturally identified with the restriction $\varphi |_\gamma$ of $\varphi$ to $\gamma$. By a trajectory of a differential $\varphi$ on $S_0$ we mean a maximal regular curve $\gamma$ such that the restriction of $\varphi$ to it is positive: $\varphi |_\gamma >0$; a curve is an orthogonal trajectory if the restriction of $\varphi$ to it satisfies the reverse inequality $\varphi |_\gamma <0$. Trajectories and orthogonal trajectories of quadratic differentials can be defined in terms of the intrinsic differential geometry of the surface and therefore are independent of the choice of a local parameter. We say that a trajectory (respectively, orthogonal trajectory) is critical if its closure contains a zero or pole of $\varphi$; otherwise it is said to be regular. For a detailed description of the local and global structure of trajectories the reader can consult [48], [103], or [118].

Teichmüller maps $f:S_0\rightarrow S$ are important for the theory of quasiconformal maps. They are generalized $K$-quasiconformal homeomorphic solutions $f$ of Beltrami's equation (1) with complex dilatation

$$\begin{equation} \mu _f(\zeta ) =k\,\frac{\overline{\varphi (\zeta )}}{|\varphi (\zeta )|},\qquad k=\frac{K-1}{K+1}, \end{equation} \tag{ 3 }$$

where $\varphi (\zeta )\,d\zeta ^2$ is a holomorphic quadratic differential on $S_0$. The inverse $f^{-1}$ of a Teichmüller map $f$ is also a Teichmüller map, that is, there exists a holomorphic quadratic differential $\psi (w)\,dw^2$ on $f(S_0)$ such that the inverse map has dilatation $\mu _{f^{-1}}(w) =k{\overline{\psi (w)}}/{|\psi (w)|}$. Teichmüller maps are locally $\mathbb {R}$-linear and take infinitesimal circles to infinitesimal ellipses with semimajor axes directed along trajectories of the differential $\psi (w)\,dw^2$ or orthogonal to these trajectories. The ratio of the semimajor axis of the ellipse to its semiminor axis is equal to $K$ at each point of $S_0$ which is not singular for $\varphi$. In other words, a quadratic differential $\varphi (\zeta )\,d\zeta ^2$ on $S_0$ has the following property: away from its zeros we can construct local coordinate charts on $S_0$ and $S$ in which $\varphi (\zeta )\,d\zeta ^2=dz^2$ and $f(x+iy)=Kx+iy$.

In the definition of the Teichmüller space of finite Riemann surfaces we follow Teichmüller's original approach.

We fix non-negative integers $g$ and $n$ and a Riemann surface $S_0$ of finite type $(g,n)$, that is, a surface of genus $g$ with $n$ punctures. If $3g-3+n>0$, then $S_0$ has hyperbolic conformal type, and its universal covering space is conformally equivalent to the unit disk $\mathbb {D}$. Deck transformations rearranging the sheets of the universal covering induce maps in the corresponding Fuchsian group $G_0$ of Möbius automorphisms of $\mathbb {D}$ which uniformizes the Riemann surface $S_0$: $S_0=\mathbb {D}/G_0$.

The Teichmüller space $T(S_0)$ is the space of analytic structures on the finite Riemann surface $S_0$, where two structures are treated as equivalent if there exists a conformal map between them which is homotopic to the identity. In other words, this is the quotient of the space of marked (with respect to the original surface $S_0$) Riemann surfaces of the given (finite) conformal type. In greater detail, let $S$ be a Riemann surface and $f$ a homeomorphism of $S_0$ onto $S$. We consider the pair $(S,[f])$, where $[f]$ is the class of homeomorphisms $S_0\rightarrow S$ homotopic to $f$. We call $(S,f)$ such that $f$ is a representative of the class $[f]$ a marked Riemann surface. Marked Riemann surfaces $(S_1,f_1)$ and $(S_2,f_2)$ are equivalent if there exists a conformal homeomorphism $h:S_1\rightarrow S_2$ such that $f_2^{-1}\circ h\circ f_1$ is homotopic to the identity map of $S_0$. A class of equivalent marked Riemann surfaces defines a point in the Teichmüller space, and the set of equivalence classes forms the Teichmüller space $T\equiv T(S_0)$ (with respect to the original surface $S_0$). The equivalence class containing the marked Riemann surface $(S_0,\operatorname{id})$ is taken to be the reference point 0 in $T(S_0)$. The Teichmüller space $T(S_0)$ is a complex analytic manifold of complex dimension $3g-3+n$.

It can also be defined as the set of equivalence classes of complex structures. Let $\varphi :U\rightarrow \mathbb {C}$ and $\psi :V\rightarrow \mathbb {C}$ be atlases corresponding to two complex structures on $S_0$. These structures are said to be equivalent if there exists a homeomorphism $f:S_0\rightarrow S_0$ isotopic to the identity such that $\varphi \circ f=\psi$. Then the space $T(S_0)$ can be identified with the set of equivalence classes of all complex structures on $S_0$.

The Teichmüller space $T(S_0)$ is a connected and simply connected metric space endowed with a natural Teichmüller metric $\tau _T(x,y)$. Let $(S_1,f_1)$ and $(S_2,f_2)$ be marked Riemann surfaces. The homeomorphism $f_2\circ f_1^{-1}$: $S_1\rightarrow S_2$ gives rise to a homeomorphism between marked Riemann surfaces. The class of homeomorphisms homotopic to $f_2\circ f_1^{-1}$ must contain quasiconformal maps (see, for instance, [16]). Assume that the marked surface $(S_1,f_1)$ corresponds to a point $x\in T(S_0)$ and $(S_2,f_2)$ corresponds to $y\in T(S_0)$. Then

$$\begin{equation*} \tau _T(x,y):=\frac{1}{2} \inf _f\log \frac{1+\Vert \mu _f\Vert _\infty }{1-\Vert \mu _f\Vert _\infty }, \end{equation*}$$

where $f$ ranges over the set of quasiconformal homeomorphisms homotopic to $f_2\circ f_1^{-1}$ and $\Vert \mu _f\Vert _\infty =\operatorname{ess}\sup _{z\in S_1}|\mu _f(z)|<1$.

Let $S_0=\mathbb {D}/G_0$ and let $B(G_0)$ denote the space of Beltrami differentials $\mu =\mu (z)\,d\bar{z}/dz$ invariant under the action of $G_0$, so that each $\mu (z)$ is a measurable function in $\mathbb {D}$ such that $\Vert \mu \Vert _\infty <\infty$ and

$$\begin{equation*} \mu (\gamma (z)) \frac{\overline{\gamma ^{\prime }(z)}}{\gamma ^{\prime }(z)} =\mu (z) \end{equation*}$$

for each $\gamma \in G_0$. Let $D(G_0)$ denote the unit ball in $B(G_0)$: $D(G_0)=\lbrace \mu \in B(G_0)$: $\Vert \mu \Vert _\infty <1\rbrace$.

Via a solution of Beltrami's equation, each differential $\mu \in D(G_0)$ determines a unique quasiconformal automorphism $\omega ^\mu :\mathbb {D}\rightarrow \mathbb {D}$ of the unit disk satisfying the normalization conditions $\omega ^\mu (\pm 1)=\pm 1$, $\omega ^\mu (i)=i$. We let $I(G_0)$ denote the space of trivial Beltrami differentials, which consists of the differentials $\mu \in D(G_0)$ such that $\omega ^\mu$ is the identity map on $\partial \mathbb {D}$. The Teichmüller space $T(S_0)$ can be identified with the quotient $T(G_0)=D(G_0)/I(G_0)$, so that we obtain the natural holomorphic projection $\Phi :D(G_0)\rightarrow T(G_0)$.

Riemann surfaces with the same conformal type $(g,n)$ give rise to isomorphic Teichmüller spaces, and therefore we can say that there exists a single Teichmüller space $T(g,n)$. However, sometimes it is more convenient to consider the Teichmüller space with a fixed reference Riemann surface $S_0$.

Let $H^{2,0}(G_0)$ be the space of holomorphic quadratic differentials $q=q(z)\,dz^2$ in the disk $\mathbb {D}$ which are $G_0$-invariant: $q(\gamma (z))(\gamma ^{\prime }(z))^2=q(z)$, $\gamma \in G_0$. In other terms, $H^{2,0}(G_0)$ is the space of parabolic 2-forms of weight $(-4)$ with respect to $G_0$ and with finite norm

$$\begin{equation*} \Vert q\Vert :=\sup _{\mathbb {D}}(1-|z|^2)^2|q(z)| <\infty . \end{equation*}$$

The space $H^{2,0}(G_0)$ projects onto $S_0$ as the space of holomorphic quadratic differentials $\varphi$ on $S_0$ which in the case of finite Riemann surfaces have a finite integral norm

$$\begin{equation*} \iint _{S_0} |\varphi (\zeta )|\,d\sigma _\zeta <\infty . \end{equation*}$$

We can define a natural pairing on $B(G_0)\times H^{2,0}(G_0)$ by

$$\begin{equation*} A_q(\mu ) =\iint _{\mathbb {D}/G_0} \mu q\,d\sigma _z. \end{equation*}$$

Let $N(G_0)$ be the space of locally trivial Beltrami differentials, that is, the subspace of $B(G_0)$ coinciding with the kernel of the operator $A_q(\mu )$. Then the tangent space to $T(G_0)$ at the initial point can be identified with $H(G_0):= B(G_0)/N(G_0)$.

We assign to it a subspace of $B(G_0)$ as follows. Let us define the Bergman integral $\Lambda _\mu :\mu \mapsto \Lambda _\mu (z)\in H^{2,0}(G_0)$ by the formula

$$\begin{equation*} \Lambda _\mu =\frac{12}{\pi }\iint _{\mathbb {D}/G_0} \frac{\overline{\mu (\zeta )}}{(\zeta -z)^4}\, d\sigma _\zeta ,\qquad \mu \in B(G_0). \end{equation*}$$

Then the space $N(G_0)$ coincides with the kernel of the operator $\Lambda _\mu$. Let $\Lambda _q^*$: $q\mapsto \Lambda _q^*(z)\in B(G_0)$ be the dual operator defined by

$$\begin{equation*} \Lambda _q^*(z) =\overline{q(z)}{(1-|z|^2)^2},\qquad q\in H^{2,0}(G_0). \end{equation*}$$

The composite $\Lambda _\mu \circ \Lambda _q^*$ is the identity operator on the space $H^{2,0}(G_0)$, and thus $\Lambda _q^*$ splits the exact sequence

$$\begin{equation*} 0\rightarrow N(G_0) \hookrightarrow B(G_0) \overset{\Lambda _\mu }{\rightarrow } H^{2,0}(G_0)\rightarrow 0. \end{equation*}$$

Hence $H(G_0)=\Lambda _q^*(H^{2,0}(G_0)) \cong B(G_0)/N(G_0)$. The tangent space $H(G)$ at other points in $T(G_0)$ can be obtained by similar arguments applied to the group $G=\chi _\mu (G_0)$, where $\chi _\mu$ is the group isomorphism defined by $\chi _\mu :G_0 \mapsto \omega ^\mu \circ G_0 \circ (\omega ^\mu )^{-1}$. The pairing $\langle \mu ,q\rangle := A(\mu )$ defines the cotangent space to be equal to $H^{2,0}(G_0)$. We have on it the Petersson Hermitian product defined by

$$\begin{equation*} (q_1,q_2) =\iint _{\mathbb {D}/G_0}q_1(z) \overline{q_2(z)}{(1-|z|^2)^2}\,d\sigma _z. \end{equation*}$$

We can introduce the Weil–Petersson Kähler metric $\lbrace \mu _1,\mu _2\rbrace =\langle \mu _1,\Lambda _{\mu _2}\rangle$ on the tangent space to $T(G_0)$ (see, for example, [124]), thereby converting $T(G_0)$ into a Kähler manifold.

As already mentioned, the real dilatation

$$\begin{equation*} K(f) =\frac{1+\Vert \mu _f\Vert _\infty }{1-\Vert \mu _f\Vert _\infty } \end{equation*}$$

measures how much a quasiconformal map $f$ deviates from being conformal. Let $(S,f)$ be a marked (with respect to $S_0$) Riemann surface and $f$ a homeomorphism $S_0\rightarrow S$ representing a homotopy class $[f]$ of homeomorphisms of $S_0$ onto $S$. A homeomorphism $g:S\rightarrow S^{\prime }$ induces a homeomorphism of marked Riemann surfaces $(S,f)\rightarrow (S^{\prime },f^{\prime })$, where $f^{\prime }=g\circ f$. Assume that $(S,f)$ and $(S^{\prime },f^{\prime })$ are Riemann surfaces of type $(g,n,l)$. As mentioned above, each homotopy class of homeomorphisms of $(S,f)$ onto $(S^{\prime },f^{\prime })$ must contain quasiconformal maps.

The following fundamental result in the theory of quasiconformal maps was stated by Teichmüller [108], [109] and proved rigorously by Ahlfors and Bers [11]. Another proof was proposed by Krushkal [58]. For convenience we split the proof into two propositions.

Proposition 1. Let $f:S_1\rightarrow S_2$ be a Teichmüller map between two marked (with respect to $S_0$) hyperbolic surfaces $S_1$ and $S_2$. Then $f$ is the unique extremal quasiconformal map in its homotopy class $[f]$. In other words, $f$ is either conformal (so that $K(f)=1$) or $K(g)>K(f)$ for any quasiconformal map $g$ homotopic to $f$ such that $f\ne g$ almost everywhere on $S_1$. In particular, $\tau _T(S_1,S_2)=(1/2)\log K(f)$.

The next proposition is often called Teichmüller's theorem.

Proposition 2. Let $S_1$ and $S_2$ be marked (with respect to $S_0$) hyperbolic surfaces in $T(S_0)$. Then there exists a unique Teichmüller map $f:S_1\rightarrow S_2$.

In particular, different Teichmüller maps can never be homotopic.

The Riemann sphere $\widehat{\mathbb {C}}$ carries a unique complex structure, so its Teichmüller space is trivial (consists of one point). The Teichmüller space of a torus $\mathbb {T}$ is isomorphic to the upper half-plane $\mathbb {H}=\lbrace z\in \mathbb {C} :\operatorname{Im}z>0\rbrace$. Teichmüller's theorem also holds in this case, that is, a Teichmüller map is defined uniquely up to conformal automorphisms of the torus $\mathbb {T}$ which are homotopic to the identity. Note that the Teichmüller spaces $T(0,4)$ and $T(1,1)$ are isometric to $T(1)$, therefore the Teichmüller metric in these cases is also hyperbolic.

Teichmüller maps enable us to identify real and complex geodesics (Teichmüller disks) in the Teichmüller metric. Namely, let $f^\mu$ be a Teichmüller map with Beltrami differential

$$\begin{equation*} \mu (z)=t\overline{\varphi (z)}/|\varphi (z)| \end{equation*}$$

on $S_0$, which assigns a point in $T(S_0)$ to $t\in \mathbb {D}$. Then the map $f:(-1,1)\rightarrow T(S_0)$ defines a real geodesic in the Teichmüller metric. On the other hand, the holomorphic embedding $f^\mu :\mathbb {D}\rightarrow T(S_0)$ defines a complex geodesic disk in $T(S_0)$.

The Teichmüller space $T(g,n)$ is a $(3g-3+n)$-dimensional hyperbolic complex analytic manifold on which we can introduce biholomorphically invariant metrics generalizing the hyperbolic metric. We are particularly interested in the Kobayashi metric $k_T(x,y)$ and the Carathéodory metric $c_T(x,y)$, $x,y\in T(g,n)$. Their definitions and properties can be found in the monographs [32], [62], and [87].

Let $S_0=\mathbb {D}/G_0$ and let $q=q(z)\,dz^2$ be a holomorphic quadratic differential in $\mathbb {D}$ satisfying the invariance condition $q(\gamma (z))(\gamma ^{\prime }(z))^2=q(z)$ for each $\gamma \in G_0$. By a Teichmüller disk we mean a complex geodesic disk in the Teichmüller metric, that is, a holomorphic embedding of the unit disk in $T(S_0)$ having the form $\Delta _q:=\lbrace \Phi (t\bar{q}/|q|),\, t\in \mathbb {D}\rbrace$, where $\Phi :D(G_0)\rightarrow T(G_0)$ is the natural holomorphic projection.

Let $d(\,\cdot \,,\,\cdot \,)$ be the standard Poincaré hyperbolic metric in the unit disk $\mathbb {D}$:

$$\begin{equation*} d(z_1,z_2)=\frac{1}{2}\log \frac{1+|(z_1-z_2)/(1-z_1\bar{z}_2)|}{1-|(z_1-z_2)/(1-z_1\bar{z}_2)|}. \end{equation*}$$

Let $X$, $A$, $B$ be complex analytic manifolds and let $\operatorname{Hol}(A,B)$ denote the class of holomorphic maps $A\rightarrow B$. The Kobayashi pseudometric $k_X(x,y)$ on $X$ is the largest pseudometric $\rho$ on $X$ satisfying $\rho (f(z_1),f(z_2))\leqslant d(z_1,z_2)$ for each $f\in \operatorname{Hol}(\mathbb {D},X)$, $z_1,z_2\in \mathbb {D}$. The Carathéodory pseudometric $c_X(x,y)$ is the smallest pseudometric $\rho$ on $X$ satisfying the inequality $\rho (x,y)\geqslant d(f(x),f(y))$ for each $f\in \operatorname{Hol}(X,\mathbb {D})$, $x,y\in X$. Both pseudometrics on $X$ have the main properties of the Poincaré hyperbolic metric: they decrease under holomorphic maps (which is an analogue of Schwarz's lemma) and are biholomorphically invariant. In particular, if $X$ is a submanifold of $T(g,n)$, then $c_X(x,y)\geqslant c_T(x,y)$. Both pseudometrics are proper metrics on the Teichmüller space itself and its submanifolds. Moreover, the Kobayashi metric is intrinsic, that is, can be recovered from its infinitesimal metric, but the Carathéodory metric does not have this property. This is perhaps the main difference between these metrics. Note that we always have $k_X(x,y)\geqslant c_X(x,y)$.

The Kobayashi metric on $T(g,0)$ was completely described by Royden [94] in 1971. In 1974 Earle and Kra (see [27] and [20]) carried over these results to $T(g,n)$: they showed that $k_T(x,y)=\tau _T(x,y)$. The metrics $k_T(x,y)$ and $c_T(x,y)$ coincide on the Teichmüller space $T(0,4)$, because it is biholomorphically equivalent to a hyperbolic domain in $\mathbb {C}$. In [20] Abikoff, Earle, and Kra stated the conjecture that these metrics coincide on any Teichmüller space. However, in 1981 Krushkal (see [59], [61] and also [62]) gave an example showing that if $3g-3+n>2$, $n\geqslant 1$, and $g>2$, then for each point $x$ in the space $T(g,n)$ there exists a $y\in T(g,n)$ such that $k_X(x,y)> c_X(x,y)$. Nevertheless, in the same year Kra [56] proved an affirmative result in this direction: he showed that both metrics coincide on Teichmüller disks $\Delta _q$ such that $q$ is the square of an Abelian holomorphic differential on $S_0$. Several results in the opposite direction were published later. In particular, it was shown that the Bers embedding in $T(g,n)$ (see [11]) is not geodesic in the Carathéodory metric (Nag [70]). Then the analogous result was established by Krushkal [60] for the universal Teichmüller space; see also [68]. For the universal Teichmüller space Shiga and Tanigawa used quasiconformal variations of the Grunsky coefficients to give an example of a non-Abelian Teichmüller disk on which the above two metrics coincide.

Let $S$ be a hyperbolic Riemann surface with finite type $(g,n)$ such that $3g-3+n>0$. Let $\operatorname{Mod}(S)$ denote the group of isotopy classes of orientation-preserving homeomorphisms $\phi :S\rightarrow S$, that is, the group of orientation-preserving homeomorphisms of $S$ modulo the homeomorphisms homotopic to the identity. The group $\operatorname{Mod}(S)$ is called the modular Teichmüller group or the mapping class group of the surface $S$. It acts on $T(g,n)$ by $\phi \cdot (S,[f]) =(S,[f\circ \phi ^{-1}])$. The quotient by this action is the Riemann moduli space $M(g,n)=T(g,n)/\operatorname{Mod}(S)$. We can say that $M(g,n)$ is obtained from $T(g,n)$ by 'forgetting the framing' on the surfaces. In particular, the modular Teichmüller group of a torus coincides with the standard modular group $\operatorname{SL}(2,\mathbb {Z})$.

If $p$ is a base point on $S$ and $\pi _1(S,p)$ the fundamental group, then the group of isotopy classes of homeomorphisms of $S$ is naturally isomorphic to the outer automorphism group $\operatorname{Aut}(\pi _1(S,p))/\pi _1(S,p)$ of $\pi _1(S,p)$. The Dehn–Nielsen theorem (see [79]) states that the modular group is an index-2 subgroup of this outer automorphism group, consisting of orientation-preserving outer automorphisms with trivial action on the second cohomology $H^2(\pi _1(S,p),\mathbb {Z}) =H^2(S,\mathbb {Z})=\mathbb {Z}$.

The action of $\operatorname{Mod}(S)$ on $T(S)$ is not free, so the moduli space is an orbifold in which precisely one point corresponds to each isomorphism class of Riemann surfaces of the type determined by $S$. The stabilizer of a surface $x\in T(S)$ is isomorphic to the conformal automorphism group of this surface.

The famous Thurston theorem [112] was first published circa 1976 as a preprint, then re-proved by Bers [19] in 1978, and finally published in 1988. It gives the following description of representatives of $\operatorname{Mod}(S)$. Each mapping class $[f]\in \operatorname{Mod}(S)$ has a representative $f$ which is periodic (that is, has finite order), or reducible, or pseudo-Anosov. A homeomorphism $f$ is reducible if there exists a set $C$ of curves on $S$ whose components are preserved by $f$, so that $f$ just permutes them. A map $f:S\rightarrow S$ is said to be pseudo-Anosov if it is a Teichmüller map with quadratic differential $\varphi$ preserved by $f$, has real dilatation $0<k<1$ and is normalized by the condition $\int _S|\varphi |=1$.

Thurston's theorem stirred up great interest in the dynamics of the action of the mapping class group $\operatorname{Mod}(S)$ on the Teichmüller space $T(S)$. Thurston constructed a compactification of $T(S)$ to which the action of $\operatorname{Mod}(S)$ extends in the natural way. The type of an element $g$ of $\operatorname{Mod}(S)$ is connected as follows with the fixed points on this compactification. If $g$ is periodic, then it has a fixed point in the interior of $T(S)$. If $g$ is pseudo-Anosov, then $g$ has no fixed points in $T(S)$, but has two fixed points on the boundary of its Thurston compactification. Some reducible maps $g$ have a unique fixed point on the Thurston boundary. This classification of homeomorphisms in the group $\operatorname{Mod}(S)$ is analogous in some sense to the classification of the Möbius automorphisms of the unit disk as transformations of elliptic, parabolic, or hyperbolic type.

There are several non-equivalent methods for compactifying the Teichmüller space $T(S_0)$.

The Teichmüller compactification consists in complementing the Teichmüller space by the limit points of geodesic rays (in the Teichmüller metric) emanating from the fixed reference point $S_0$. This compactification depends on the choice of the reference point and thus cannot be projected onto the moduli space. Moreover, Kerckhoff [52] showed that the action of the modular group on the Teichmüller space does not extend to a continuous action of it on this compactification.

In [18] Bers constructed a compactification coinciding with the closure of $T(g)$ regarded as a bounded subdomain of $\mathbb {C}^{3g-3}$. We construct a map $\Psi :B(G_0)\rightarrow H^{2,0}(G_0)$ as follows. The Beltrami differential $m(z)\in B(G_0)$ extends by symmetry to the whole plane: $m(1/\bar{z})=\bar{m}(z)z^2/\bar{z}^2$, $z\in \widehat{\mathbb {C}} \setminus \overline{\mathbb {D}}$. We set

$$\begin{equation*} \mu (z)=\left\lbrace \begin{array}{@{}l@{\quad }l@{}}0, &z\in \mathbb {D}, \\ m(z), &z\in \widehat{\mathbb {C}} \setminus \overline{\mathbb {D}}. \end{array}\right. \end{equation*}$$

Let $f^\mu$ be the solution of Beltrami's equation $w_{\bar{z}}=\mu (z)w_z$ which is conformal in $\mathbb {D}$ and normalized by the condition $f(0)=f^{\prime }(0)-1=0$. Nehari [76] and Kraus [57] showed that $\Vert S_{f^\mu }\Vert \leqslant 6$, where

$$\begin{equation*} S_f=\frac{f^{\prime \prime \prime }(z)}{f^{\prime }(z)} -\frac{3}{2} \biggl (\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\biggr )^2,\qquad z\in \mathbb {D}, \end{equation*}$$

is the Schwarzian derivative. Note that the inequality $\Vert S_{f}\Vert \leqslant 2$ ensures that $f$ can be recovered from $S_f$. We set $\Psi (\mu )=S[f^\mu |_{\mathbb {D}}]$. Then $\Psi$ defines an injective holomorphic map of the Teichmüller space $T(G_0)$ onto an open subset of $H^{2,0}(G_0)$. Its image contains a ball of radius 2 with centre at zero and lies in the concentric ball with radius 6. The action of the modular group cannot be continuously extended to the Bers compactification either [53].

Only the Thurston compactification [112] is compatible with the action of the modular group. The main tool for its construction is a measured foliation on a given surface. A foliation is an integrable subbundle of the tangent bundle. Locally, it defines a subdivision of the manifold into parallel submanifolds of lower dimension. A measured foliation consists of a foliation (which can be singular in the general case) and a measure 'concentrated' in the direction transversal to the foliation. The Thurston compactification coincides with the compactification of the space $\operatorname{PMF}(S_0)$ of projective measured foliations.

Let $C$ be the set of isotopy classes of simple closed curves on $S_0$ and $R^+(C)$ the set of positive functions on $C$, with the product topology. Let $\mathbb {P}R^+(C)$ denote the corresponding projective space. The space $T(S_0)$ can be regarded as the space of hyperbolic structures $(S_0,\rho )$ on $S_0$ modulo the following equivalence relation: two structures are equivalent, $(S_0,\rho _1)\sim (S_0,\rho _2)$, if there exists an isometry $(S_0,\rho _1)\rightarrow (S_0,\rho _2)$ isotopic to the identity map. So it is possible to construct a map $T(S_0)\rightarrow \mathbb {P}R^+(C)$. The image of the space $T(S_0)$ in $\mathbb {P}R^+(C)$ is homeomorphic to the open ball in $\mathbb {R}^{6g-6}$, and its boundary $\operatorname{PMF}(S_0)$ is a ($6g-7$)-sphere. The modular group acts continuously on this Thurston compactification, and each element of it has a fixed point on the compactification.

Now we return to the holomorphic covering $P:T(G_0)\rightarrow M(G_0)$. The moduli space $M(G_0)$ has the well-known Deligne–Mumford compactification $\widehat{M}(G_0)$ (see [25]) obtained by adding stable Riemann surfaces to $M(G_0)$. These are surfaces obtained from ordinary Riemann surfaces by 'constricting' (that is, contracting simple closed geodesics to a point). It is desirable to lift this compactification to the Teichmüller space, or in other words, to understand the (perhaps partial) completion of $T(G_0)$ corresponding to this compactification of the moduli space. Such a completion $NT(G_0)$, called the nodal Teichmüller space or Teichmüller space with nodes, was constructed in [41]. The map $P$ projects it onto the Deligne–Mumford compactification of the moduli space. We note that the construction in [41] is also suitable for Kleinian groups.

If $G_0$ is a Fuchsian group of the first kind, then the above partial completion coincides with the augmented Teichmüller space constructed by Abikoff [1], [2]. There is also a paper by Masur [69], who investigated the completion of the Weil–Petersson metric and interpreted the boundary points of this completion as nodal Riemann surfaces. More precisely, Masur identified the Weil–Petersson completion with Abikoff's augmented space, whose boundary is a stratified union of spaces of lower dimension, each being the Teichmüller space of a nodal surface. He showed that in a certain precise sense the tangential component of the Weil–Petersson metric on the Teichmüller space extends to the Weil–Petersson metric on these augmented Teichmüller spaces.

For more information about compactifications see [46].

The notion of the modulus of a family of curves, which is converse to the Ahlfors–Bers notion of extremal length, underlies the geometric definition of quasiconformal maps. It was proposed by Grötzsch [38] in 1928 and developed further by Ahlfors and Bers [12] in 1946. Since then several surveys and monographs on this subject have been published, for instance, [26], [48], [63], [83], [102], [118].

In this subsection, we use the modulus of a free family of homotopy classes of curves on the Riemann surface $S_0$ to construct a functional called the modulus on the Teichmüller space $T(S_0)$. Then we discuss the harmonic properties of this functional and describe the Teichmüller metric in terms of it. These results were expounded in [114] and [118].

Let $\Gamma$ be a family of curves on the surface $S_0$ and let $\rho$ denote a metric on $S_0$ defined locally by the form $\rho (\zeta )\,|d\zeta |$. In other words, we assign a non-negative real measurable function $\rho (\zeta )$ satisfying the following invariance condition to an arbitrary local parameter $\zeta$ on $S_0$:

$$\begin{equation*} \rho ^*(\zeta ^*)=\rho (\zeta )\, \biggl |\frac{d\zeta }{d\zeta ^*}\biggr |. \end{equation*}$$

We fix a system of curves $\lbrace \gamma _1,\dots ,\gamma _m\rbrace$ on a Riemann surface $S_0$ of finite type $(g,n,m)$ which we call curves of types I and II. The first family in it (curves of type I) consists of simple loops on $S_0$ which are pairwise freely non-homotopic to one another. The second family (curves of type II) exists only when $S_0$ has hyperbolic boundary components (that is, for $l>0$) and consists of simple arcs on $S_0$ which have endpoints on these components. Here we assume that all the curves are homotopically non-trivial (that is, not homotopic to a point on the surface or its closure) and pairwise disjoint. We call such families of curves admissible systems.

We say that a set $\Gamma _j$ of curves on $S_0$ forms the homotopy class generated by some curve $\gamma _j$ in an admissible system $\lbrace \gamma _1,\dots ,\gamma _m\rbrace$ if this set consists of all the curves of type I (or type II) that are freely homotopic to $\gamma _j$ on $S_0$. Let $\alpha :=(\alpha _1,\dots ,\alpha _m)$ be a non-zero vector with non-negative components. Let $P$ be the family of metrics $\rho$ on $S_0$ satisfying the following admissibility condition (see [47] or [105]):

$$\begin{equation} \int _\gamma \rho (\zeta )\,|d\zeta | \geqslant \alpha _j \end{equation} \tag{ 4 }$$

for any $\gamma \in \Gamma _j$, $j=1,\dots ,m$. (Following [105], we use in (4) the lower Darboux integral to avoid integration along non-rectifiable curves, which can occur when quasiconformal maps are applied to rectifiable curves.)

If $P\ne \varnothing$, then we say that the modulus problem is well posed for the given family $\Gamma =\lbrace \Gamma _1,\dots ,\Gamma _m\rbrace$ on $S_0$ and the vector $\alpha$; by definition its solution, called the modulus, is defined by

$$\begin{equation} m(S_0,\Gamma ,\alpha ) =\inf _{\rho \in P}\iint _{S_0} \rho ^2(\zeta )\,d\sigma _\zeta . \end{equation} \tag{ 5 }$$

It is known (see [47]) that the extremal metric $\rho ^*$ in the modulus problem stated above is uniquely defined. It corresponds to a unique holomorphic quadratic differential $\varphi =\varphi (\zeta )\,d\zeta ^2$ in $H^{2,0}(S_0)$ which has finite trajectories on $S_0$; namely, $\rho ^*(\zeta )=\sqrt{|\varphi (\zeta )|}$. It is clear from the definition of the modulus that its value is a conformal invariant.

We say that a hyperbolic doubly connected domain $D_j\subset S_0$ on the surface is associated with a homotopy class $\Gamma _j$ of curves of type I if any simple loop separating the boundary components of $D_j$ belongs to $\Gamma _j$. We can give a similar definition also for curves of type II. Namely, let $D_j$ be a quadrilateral with pairs of different vertices 1, 4 and 2, 3 lying on two different hyperbolic components of the boundary of $S_0$. We say that $D_j$ is associated with a homotopy class $\Gamma _j$ of curves of type II if $\Gamma _j$ contains any simple arc that connects the boundary component containing 1 and 4 with the boundary component containing 2 and 3.

The critical trajectories of the differential $\varphi =\varphi (\zeta )\,d\zeta ^2$ decompose $S_0$ into at most $m$ annular domains and quadrilaterals associated with the corresponding homotopy classes of curves. Let $\mathfrak {D}^*=\lbrace D^*_1,\dots , D^*_m\rbrace$ be the system of these domains (we note that some of the quadrilaterals in this system can be degenerate). By analogy with families of curves we call the doubly connected domains in this system domains of type I, and the quadrilaterals will be domains of type II. Any system $\mathfrak {D}=\lbrace D_1,\dots , D_m\rbrace$ of $m$ disjoint doubly connected domains and quadrilaterals of types I and II associated with the homotopy classes in the family $\Gamma$ satisfies the inequality

$$\begin{equation*} \sum _{j=1}^m\alpha _j^2M(D_j) \leqslant \sum _{j=1}^m\alpha _j^2M(D_j^*), \end{equation*}$$

with equality only for $\mathfrak {D}=\mathfrak {D}^*$. Here $M(D_j)$ is the modulus of the doubly connected domain with respect to the family of curves separating its boundary components or the modulus of a quadrilateral with respect to the family of curves connecting 'opposite' hyperbolic components of the boundary $\partial S_0$.

As we have noted, each domain $D_j^*$ is an annular domain or a quadrilateral drawn by trajectories of the differential $\varphi =\varphi (\zeta )\,d\zeta ^2$. If it is an annular domain, then there exists a conformal map $g_j$ of $D_j^*$ which solves the differential equation

$$\begin{equation*} \alpha _j^2 \biggl (\frac{g^{\prime }_j(\zeta )}{g_j(\zeta )}\biggr )^2 =-4\pi ^2\varphi (\zeta ). \end{equation*}$$

It maps $D_j^*$ onto the annulus $\lbrace 1<|w|<\exp (2\pi M(D_j^*))\rbrace$. If $D_j^*$ is a quadrilateral, then there exists a conformal map $g_j$ of $D_j^*$ satisfying the differential equation

$$\begin{equation*} \alpha _j^2(g^{\prime }_j(\zeta ))^2 =\varphi (\zeta ). \end{equation*}$$

It maps $D_j^*$ onto the rectangle $\lbrace 0<\operatorname{Re}w<1,\, 0<\operatorname{Im}w<M(D_j^*)\rbrace$. We call the system of domains $\mathfrak {D}^*$ the characteristic system of domains for the quadratic differential $\varphi =\varphi (\zeta )\,d\zeta ^2$.

Then the problem of finding the modulus for a free family of homotopy classes of curves is equivalent to constructing an extremal decomposition of the Riemann surface $S_0$ into domains of the indicated types associated with the given family.

On a Riemann surface $S_0$ of type $(g,n,l)$ we now pick an admissible system $\lbrace \gamma _1,\dots ,\gamma _m\rbrace$ of curves of types I and II generating the corresponding free family $\Gamma =\lbrace \Gamma _1,\dots ,\Gamma _m\rbrace$ of homotopy classes of curves. For the free family $\Gamma$ and a fixed weight vector $\alpha$ we consider the modulus problem (4), (5) and denote the modulus in question by $m_0=m(S_0,\Gamma ,\alpha )$. Assume that a marked Riemann surface $(S,f)$ corresponds to a point $x$ in the Teichmüller space $T(S_0)$. The corresponding quasiconformal map $w=f(\zeta ):S_0\rightarrow S$ has complex dilatation $\mu _f=f_{\bar{\zeta }}/f_\zeta$. Then we can look at the modulus problem (4), (5) on the Riemann surface $S$ for the admissible system of curves $\lbrace f(\gamma _1),\dots ,f(\gamma _m)\rbrace$ and the same weight vector $\alpha$. The corresponding free family of curves $\Gamma _f$ on $S$ has a modulus $m_f=m(S,\Gamma _f,\alpha )$. If $\varphi _\mu (w)\,dw^2$ (where $w$ is a local parameter on $S$) is the extremal quadratic differential for the modulus problem on $S$, then let

$$\begin{equation*} m_f=\iint _S|\varphi _\mu |\,d\sigma _w. \end{equation*}$$

Thus, we have assigned a certain functional $m_f$ to marked Riemann surfaces $(S,f)$, which we shall also call the modulus. Since the homotopy type of curves on $S$ is uniquely determined by the homotopy types of maps in the same equivalence class $x\in T(S_0)$, it follows from the conformal invariance of the modulus that the value of $m_f$ only depends on the point $x\in T(S_0)$, but not on the choice of the particular map $f$. Therefore, in what follows we denote the modulus $m_f$ by $m(x)$, where $x$ ranges over the space $T(S_0)$ while the admissible system $\Gamma$, the weight vector $\alpha$, and the initial condition $m(0)=m_0$ are fixed. If $\Phi (\mu _1)=\Phi (\mu _2)=x$, then the differential $\varphi _{\mu _1}$ can be obtained from $\varphi _{\mu _2}$ by a conformal transformation. Hence, $\varphi _{\mu _1}$ and $\varphi _{\mu _2}$ represent the same quadratic differential $\varphi _x$ with $\varphi _0\equiv \varphi$.

We are interested in the dependence of the modulus $m(x)$ on the point $x$ in $T(S_0)$. Let

$$\begin{equation} t=\inf \Vert \mu \Vert _\infty , \end{equation} \tag{ 6 }$$

where the infimum is taken over all $\mu \in B(G_0)$ such that $\Phi (\mu )=x\in T(G_0)$. Let $\mu ^*\in B(G_0)$ be an extremal Beltrami differential giving the infimum in (6): $\Vert \mu ^*\Vert _\infty =t$.

Proposition 3. [117]. The following formula holds:

$$\begin{equation*} m(x)=m(0)- 2\operatorname{Re} \iint _{\mathbb {D}/G_0}\mu ^*(z)q(z)\, d\sigma _z+o(t), \end{equation*}$$

where $q=q(z)\,dz^2$ is the lift of the extremal quadratic differential $\varphi$ to the universal covering space $\mathbb {D}$.

We now investigate the local harmonic properties of the modulus $m(x)$ on the Teichmüller space $T(S_0)$ of the original Riemann surface $S_0$. We assume that this is a surface of genus $g$ with $n$ punctures such that $3g+n-3>0$.

Proposition 4. [118]. Let $m= 3g-3+n$. Fix an admissible system $\Gamma$ of $m$ curves and a weight vector $\alpha =(\alpha _1,\dots ,\alpha _m)$ with positive components $\alpha _j$ and let $m(S_0,\Gamma ,\alpha )$ be its modulus. Assume that the trajectories of the extremal quadratic differential $\varphi _0$ decompose $S_0$ into $m$ non-degenerate annular domains. Then the reference point in the Teichmüller space $T(S_0)$ has a neighbourhood $\Omega$ such that the restriction of the modulus $m(x)$ to this neighbourhood is a harmonic function.

Simple counterexamples show that $m(x)$ is not a pluriharmonic function. However, the following statement holds.

Proposition 5 [118]. For an arbitrary admissible system $\Gamma$ and any weight vector $\alpha$ the extension $m(x)$ of the modulus $m_0=m(S_0,\Gamma ,\alpha )$ is a subharmonic function on the whole Teichmüller space $T(S_0)$.

The Teichmüller metric can also be defined in terms of moduli. Let $m(S_0,\Gamma ,\alpha )$ be the modulus of a given admissible system $\Gamma$ of curves of types I and II on $S_0$ with weight vector $\alpha$, and let $m(x)$ denote the extension of this modulus to the Teichmüller space $T(S_0)$. Then the following result holds.

Proposition 6 [115]. The Teichmüller metric $\tau _T(x,y)$ in the Teichmüller space $T(S_0)$ can be described as follows:

$$\begin{equation*} \tau _T(x,y)=\sup _{\Gamma ,\alpha } \frac{1}{2} \biggl |\log \frac{m(x)}{m(y)}\biggr | =\sup _{\Gamma ,\alpha }\frac{1}{2} \biggl |\log \frac{m(f_1(S_0),f_1(\Gamma ),\alpha )}{m(f_2(S_0),f_2(\Gamma ),\alpha )}\biggr |, \end{equation*}$$

where $(f_1(S_0),f_1)$ and $(f_2(S_0),f_2)$ are marked Riemann surfaces corresponding to the points $x$ and $y$, respectively.

A similar description in terms of the Ahlfors–Beurling extremal length was given by Kerckhoff [52] for compact and punctured Riemann surfaces.

The following theorem is based on the harmonic properties of the functional $m(x)$ and gives a condition ensuring that Teichmüller disks or their subdisks are geodesic in the Carathéodory metric (see § 2.5).

Theorem 1 [116]. Let $X_r=\lbrace x:x\in T(g,n),\,\tau _T(0,x)<r\rbrace$. Let $\varphi$ be the extremal quadratic differential for the functional $m(x)$ at $x=0$, and let $q$ be its lift to the universal covering space $U$. If $m(x)$ is pluriharmonic on $X_r$, then the Teichmüller– Kobayashi metric $\tau _T$ and the Carathéodory metric $c_{X_r}$ coincide on the disk $\Delta _q\cap X_r$. In particular, this holds for the whole of $T(g,n)$ since $X_r\rightarrow T(g,n)$ as $r\rightarrow \infty$.

The modulus problem (4), (5) is closely connected with the Dirichlet principle. Namely, fix a Riemann surface $S_0$ of finite type $(g,n,l)$ and a free family of homotopy classes of curves of types I and II on $S_0$. The Dirichlet problem consists in finding the minimum of the $L_1$-norm of the quadratic differential $\varphi$ corresponding to the complex structure on $S_0$ among all the differentials of a given homotopy type. The length of the trajectories of this differential in the metric $\sqrt{|\varphi |}$ does not exceed an explicitly defined quantity. The Dirichlet principle asserts that there exists a unique differential giving this minimum. In this form the Dirichlet principle is a special case of the modulus problem when only metrics of the form $\sqrt{|\varphi |}$ are taken in the class of admissible metrics. Thus, it comes as little surprise that the solutions of both problems give rise to the same metric (see [48] and [103]).

A closely related approach was used by Thurston (see the collection of papers [29] based on Thurston's 1976–1977 Orsay lectures). His approach was also based on the Dirichlet principle. In greater detail: one introduces measured foliations in the space of homotopy classes of curves and seeks the minimum of the $L^1$-norm in the space of differentials of curves of a given homotopy type with a prescribed bound for the height in the metric $|\operatorname{Im}\sqrt{\varphi }|$. This foliation is constructed by formal multiplication of the homotopy classes by positive numbers, so that each class corresponds to a separate ray of the foliation.

The norm of a measured foliation plays a role analogous to that of the norm of the extremal quadratic differential in the modulus problem. Using this norm, one can describe the Teichmüller metric and investigate the harmonic properties of the norm and its connections with the Kobayashi and Carathéodory invariant metrics just as in the modulus problem.

In Newton's classical mechanics a point mass moves along the shortest trajectory (geodesic) between its initial and final position in the absence of external forces. When we pass to the quantum case, this changes radically. In accordance with Feynman's idea of 1948, an elementary particle can move along any trajectory connecting the initial and final positions. More precisely, to each trajectory we can assign some value of a probability density, which is greater for 'shorter' trajectories, or, in the Lagrangian setting, for smaller action. Here particles are still regarded as topological points, but in quantum mechanics they can also have some additional intrinsic structure.

A string is a 1-dimensional object moving in a flat $D$-dimensional space-time $\mathfrak {M}$, where the first coordinate is identified with time. In its motion the string sweeps out a 2-dimensional surface $M$ in $\mathfrak {M}$ called the world sheet of the string, by analogy with the world line of a Newtonian particle. We wish to define a probability in the space of surfaces connecting the initial and final 'positions' of the string with the following property: this probability must be greater for smaller action.

Denoting the action by $S[\phi ]$, where $\phi$ is a quantity characterizing the surface (this will be the metric in what follows), we note that the Gaussian distribution yields the simplest Feynman path integral

$$\begin{equation} Z=\int _{\mathrm{ini}}^{\mathrm{fin}}D\phi \, e^{-(1/\hbar )S[\phi ]}, \end{equation} \tag{ 7 }$$

or statistical sum (partition function), where $\hbar$ is Planck's constant and $D\phi$ plays the role of a measure and is, roughly speaking, a local product $\prod _{x\in \mathfrak {M}}\,d\mu (\phi (x))$ of measures on $\mathfrak {M}$. The symbol $\int D\phi$ is the short notation for an infinite-dimensional integral over all possible field configurations on the space-time; strictly speaking, it is not a rigorously defined mathematical object. The reason is that there is no translation-invariant measure on an infinite-dimensional space. Thus, this path integral must be treated as some formal symbol, which only acquires a precise mathematical meaning in some special cases. For instance, in dimension $D=26$ it can be consistently defined as an integral over the Teichmüller space with respect to the Weil–Petersson measure.

We parametrize the world sheet of the string $M$ by parameters $(\tau ,\sigma )$ and let $x^\mu$, $\mu =0,1,\dots ,D-1$, denote the coordinate variables in the space-time $\mathfrak {M}$. While the parameters $(\tau ,\sigma )$ have no straightforward physical meaning, the functions $x^{\mu }(\tau ,\sigma )$, which describe the position of the point on the world sheet, characterize the motion of the string in the space-time $\mathfrak {M}$ and therefore are called dynamical variables.

A string is said to be closed if the dynamical variables are periodic functions of $\sigma$, otherwise it is open. The parameter $\tau$ ranges over the whole line $(-\infty ,\infty )$ or over the half-line $[0,\infty )$. In what follows we limit ourselves to closed strings.

Let $\mathfrak {M}$ be a $D$-dimensional vector space endowed with a Lorentz metric $ds_{\mathfrak {M}}$ with signature $(1,D-1)$, so that

$$\begin{equation*} ds^2_{\mathfrak {M}} =\eta _{\mu \nu }\,dx^\mu \,dx^\nu ,\qquad \eta _{\mu \nu }={\left(\begin{array}{*{10}c}-1 &0 &\dots &0 \\ 0 &1 &\dots &0 \\ \dots &\dots &\dots &\dots \\ 0 &0 &\dots &1 \end{array}\right)}. \end{equation*}$$

We consider the following simplified model of a closed bosonic string. Let $x^\mu (\sigma ^\alpha )$ be the dynamical variables, where $\mu =0,1,\dots ,D-1$ and $\alpha =0,1$. The coordinate variables $\sigma ^0=\tau$ and $\sigma ^1=\sigma$ are defined on the canonical cylinder $C=\lbrace \tau \in (-\infty ,\infty )$, $\sigma \in [0,2\pi )\rbrace$, with the intrinsic metric

$$\begin{equation*} ds^2_C =\eta _{\alpha \beta }\,d\sigma ^\alpha \,d\sigma ^\beta ,\qquad \eta _{\alpha \beta }={\left(\begin{array}{*{10}c}-1 &0 \\ 0 &1 \end{array}\right)}. \end{equation*}$$

We assume that the dynamical variables define a $C^\infty$-embedding of $C$ in the space $\mathfrak {M}$, so that $M$ can be regarded as a smooth manifold embedded in $\mathfrak {M}$. The induced metric on the world sheet $ds_{\mathrm{ws}}$ has the form

$$\begin{equation*} ds^2_{\mathrm{ws}} =\eta _{\mu \nu }\, \frac{\partial x^\mu }{\partial \sigma ^\alpha }\, \frac{\partial x^\nu }{\partial \sigma ^\beta }\, d\sigma ^\alpha \,d\sigma ^\beta =g_{\alpha \beta }\,d\sigma ^\alpha \,d\sigma ^\beta . \end{equation*}$$

The dynamical variables $x^{\mu }(\tau ,\sigma )$ satisfy the equations of motion following from the principle of least action.

In Lagrangian mechanics a physical system is characterized by the configuration space $\mathfrak {N}$ and the Lagrangian $L$, which is a smooth real function on the Cartesian product $T\mathfrak {N}\times \mathbb {R}$ of the tangent bundle $T\mathfrak {N}$ and the real axis $\mathbb {R}$. Let $P\mathfrak {N}$ be the trajectory space, the space of smooth paths with initial point ('ini') and endpoint ('end') fixed. This space has the structure of an infinite-dimensional Fréchet manifold. The action functional $S$ is the real-valued integral functional defined by $S=\int _{\mathrm{ini}}^{\mathrm{fin}}L\,dt$, where $t$ is the real parameter on the curve. In classical mechanics the action is the difference between the kinetic and potential energy. There is no direct analogue of energy in relativistic mechanics, so Nambu [75] in 1970 and Gotô [36] in 1971 proposed taking the area of the world sheet as the simplest action. Known as the Nambu–Gotô action, this is defined by the formula

$$\begin{equation*} S_{\mathrm{NG}}=-T\int _M\,dx =-T\int _Cd\sigma ^2\,\sqrt{|\det g_{\alpha \beta }|}, \end{equation*}$$

where $C$ is the canonical cylinder and the quantity $T$, called the string tension, has the dimension of mass times length. (We note that the minus sign before the integral comes from the Lorentz metric.)

The geodesic motion of the string is described by dynamical variables satisfying the following Euler–Lagrange equations:

$$\begin{equation} \frac{1}{\sqrt{|\det g_{\alpha \beta }|}}\, \frac{\partial }{\partial \sigma ^{\alpha }} \biggl (\sqrt{|\det g_{\alpha \beta }|}\, g^{\alpha \beta }\, \frac{\partial x^\mu }{\partial \sigma ^\beta }\biggr )=0,\qquad \mu =0,\dots ,D-1, \end{equation} \tag{ 8 }$$

where the indices in $g^{\alpha \beta }$ and $g_{\alpha \beta }$ are raised and lowered with the help of the intrinsic Lorentz metric. The metric $g_{\alpha \beta }$ is defined on the world sheet $M$ embedded in $\mathfrak {M}$ and depends on the variables $x^\mu$.

Note that the Euler–Lagrange equations (8) for the dynamical variables $x^\mu$ are strongly non-linear, and the presence of a square root under the integral sign leads to extra complications in quantization. To overcome these, Polyakov [88] proposed in 1981 a method similar to the method of Lagrange multipliers. As an additional variable he introduced a metric $ds^2=h_{\alpha \beta }(\tau ,\sigma )\, d\sigma ^\alpha \,d\sigma ^\beta$ on the world sheet, so that the action took the form

$$\begin{equation*} S_{\mathrm{P}}=-\frac{T}{2} \int _Cd\sigma ^2\, \sqrt{|\det h_{\alpha \beta }|}\, h^{\alpha \beta }\eta _{\mu \nu }\, \frac{\partial x^\mu }{\partial \sigma ^\alpha }\, \frac{\partial x^\nu }{\partial \sigma ^\beta }. \end{equation*}$$

For the Polyakov action $S_{\mathrm{P}}$ the Euler–Lagrange equations (with respect to the variation of the dynamical variables) have apparently the same form as for the Nambu–Gotô action $S_{\mathrm{NG}}$:

$$\begin{equation} \frac{1}{\sqrt{|\det h_{\alpha \beta }|}}\, \frac{\partial }{\partial \sigma ^\alpha } \biggl (\sqrt{|\det h_{\alpha \beta }|}\, h^{\alpha \beta }\, \frac{\partial x^\mu }{\partial \sigma ^\beta }\biggr )=0, \end{equation} \tag{ 9 }$$

but they have another meaning, because the metric $h=(h_{\alpha \beta }(\tau ,\sigma ))$ only depends on $(\tau ,\sigma )$ but not on $x^\mu$. Therefore, equations (9) are linear in $x^\mu (\tau ,\sigma )$.

However, we shall pay for having introduced the additional metric $h=(h_{\alpha \beta })$: our system gets three additional degrees of freedom (just three because of the symmetry of the metric $h$). We can eliminate these by means of the following equations of motion for $h$:

$$\begin{equation} \frac{\delta S_{\mathrm{P}}}{\delta h^{\alpha \beta }}=0, \end{equation} \tag{ 10 }$$

where the left-hand side contains a functional derivative. From (9) and (10) we deduce necessary conditions for the action to have a minimum. Namely, we define the 2-dimensional energy-momentum tensor by the formula

$$\begin{equation*} T_{\alpha \beta }=\frac{-2}{T}\, \frac{1}{\sqrt{|\det h_{\alpha \beta }|}}\, \frac{\delta S_{\mathrm{P}}}{\delta h^{\alpha \beta }}. \end{equation*}$$

Then it follows from the equations of motion that $T_{\alpha \beta }=0$. Note that $S_\mathrm{P}=S_{\mathrm{NG}}$ under this condition (whereas $S_\mathrm{P}\geqslant S_{\mathrm{NG}}$ in the general case).

According to one of the guiding physical principles, the so-called 'Noether theorem', symmetries of the space-time give rise to conservation laws. (This statement must be treated as a heuristic principle rather than a theorem, since Noether's theorem as such, which she proved in 1918 (see [78]), relates to a very special case of the indicated principle.)

The symmetries of the Polyakov action $S_{\mathrm{P}}$, that is, the transformations preserving $S_{\mathrm{P}}$, are as follows.

• Global symmetries:
  • –  
    Poincaré transformations

    $$\begin{equation*} x^\mu \rightarrow x^\mu +b^\mu ,\qquad x^\mu \rightarrow x^\mu +\omega ^\mu _\nu x^\nu , \end{equation*}$$

    where $\omega ^\mu _\nu =-\omega ^\nu _\mu$ are infinitesimal Lorentz transformations.
• Local symmetries:
  • –  
    reparametrization transformations defined by area-preserving 2D-diffeomorphisms $\sigma ^\alpha \rightarrow \tilde{\sigma }^\alpha (\tau ,\sigma )$,

    $$\begin{equation*} d\tilde{\sigma }^2\sqrt{|\det \tilde{h}|} =d\sigma ^2\sqrt{|\det h|}; \end{equation*}$$
  • –  
    Weil scaling

    $$\begin{equation*} h_{\alpha \beta }\,d\sigma ^\alpha \, d\sigma ^\beta \rightarrow e^{\rho (\tau ,\sigma )} h_{\alpha \beta }\,d\sigma ^\alpha \, d\sigma ^\beta , \end{equation*}$$

    which leaves invariant the product $\sqrt{|\det h|}\,h_{\alpha \beta }$.

The invariance of $S_{\mathrm{P}}$ under Poincaré transformations and reparametrization allows us to choose the scaling such that the three independent components of the metric $h$ can be expressed in terms of a single function: $h_{\alpha \beta }=e^{\rho (\tau ,\sigma )}\eta _{\alpha \beta }$. This is usually identified as the conformally flat scaling. Substituting this expression in the Polyakov action, we get that

$$\begin{equation*} S_{\mathrm{P}}=-\frac{T}{2} \int _Cd\sigma ^2\, \eta ^{\alpha \beta }\eta _{\mu \nu }\, \frac{\partial x^\mu }{\partial \sigma ^\alpha }\, \frac{\partial x^\nu }{\partial \sigma ^\beta }, \end{equation*}$$

hence for a fixed scaling the action functional is quadratic in $x$. Taking its variation with respect to $x$ we obtain the equation of motion for the free field

$$\begin{equation} \ddot{x}^\mu -(x^\mu )^{\prime \prime }=0, \end{equation} \tag{ 11 }$$

where $\dot{x}$ denotes the $\tau$- and $x^{\prime }$ the $\sigma$-derivative.

It follows from the Weyl invariance that $S_{\mathrm{P}}$ is independent of $\rho (\tau ,\sigma )$. Therefore, the variation $\delta S_\mathrm{P}/\delta \rho =h^{\alpha \beta }T_{\alpha \beta } =T_\alpha ^\alpha$ vanishes, and the tensor $T_{\alpha \beta }$ must be traceless.

Using the relation $T_{\alpha \alpha }=0$, we find that $T_{01}=T_{10}=\dot{x}_\mu (x_\mu )^{\prime }=0$ and

$$\begin{equation} T_{00}=T_{11} =\frac{1}{2}(\dot{x}_\mu \dot{x}^\mu +x^{\prime }_\mu (x^\mu )^{\prime })=0. \end{equation} \tag{ 12 }$$

This gives us the equations $(\dot{x}\pm x^{\prime })^2=0$, called the Virasoro constraints. Equation (11) with the constraints (12) (and the boundary conditions) completely describes the motion of a bosonic string.

Now we use the periodic boundary conditions $x^\mu (\tau ,\sigma )=x^\mu (\tau ,\sigma +2\pi )$. Then the general solution of the wave equation of motion (11) has the form

$$\begin{equation*} x^\mu (\tau ,\sigma ) =x^\mu _{\mathrm{R}}(\tau -\sigma ) +x^\mu _{\mathrm{L}}(\tau +\sigma ). \end{equation*}$$

We assert that the centre of mass of the world sheet moves as a free particle. Using the boundary conditions and separating out the linear part, we represent the solution by Fourier series of the following form:

$$\begin{equation*} \begin{aligned} x^\mu _{\mathrm{R}}(\tau -\sigma ) &=\frac{1}{2}x_0^\mu +\frac{1}{2\pi T}(\tau -\sigma )p^\mu +\frac{i}{\sqrt{2\pi T}} \sum _{n\ne 0}\frac{1}{n} \alpha _n^\mu e^{-in(\tau -\sigma )}, \\ x^\mu _{\mathrm{L}}(\tau +\sigma ) &=\frac{1}{2}x_0^\mu +\frac{1}{2\pi T}(\tau +\sigma )p^\mu +\frac{i}{\sqrt{2\pi T}}\sum _{n\ne 0} \frac{1}{n}\beta _n^\mu e^{-in(\tau +\sigma )}, \end{aligned} \end{equation*}$$

where $x^\mu _0$ is the centre of mass and $p^\mu$ is the momentum. The functions $x^\mu _{\mathrm{R}}$ and $x^\mu _{\mathrm{L}}$ are real-valued, so that $\bar{\alpha }^\mu _n=\alpha ^\mu _{-n}$ and $\bar{\beta }^\mu _n=\beta ^\mu _{-n}$. Physicists call these coefficients oscillators.

The position of the centre of mass of the string is described by the formula

$$\begin{equation*} X^\mu =\frac{1}{2\pi } \int _0^{2\pi }x^\mu (\tau ,\sigma )\,d\sigma =x_0^\mu +\frac{p^{\mu }}{\pi T}\tau , \end{equation*}$$

which shows that the centre of mass moves like a free particle with initial position $x_0^\mu$. The momentum of the centre of mass can be expressed as

$$\begin{equation*} P^\mu =\int _0^{2\pi }\Pi ^\mu \,d\sigma =p^\mu , \end{equation*}$$

where $\Pi ^\mu :=(T/2)\dot{x}^\mu$.

In classical Hamiltonian mechanics the action $S$ is given by the integral $S=\int _{t_0}^{t_1}L(x,\dot{x})\,dt$ of the Lagrange function with respect to time. In our case the Lagrange function has the form

$$\begin{equation*} L=-\frac{T}{2}\int _0^{2\pi } \eta ^{\alpha \beta }\eta _{\mu \nu }\, \frac{\partial x^\mu }{\partial \sigma ^\alpha }\, \frac{\partial x^\nu }{\partial \sigma ^\beta }\,d\sigma . \end{equation*}$$

The Hamiltonian is

$$\begin{equation*} H=\int _0^{2\pi }(\dot{x}^\mu \Pi _\mu )\,d\sigma -L =\frac{T}{2}\int _0^{2\pi }(\dot{x}^\mu \dot{x}_\mu +(x^\mu )^{\prime }x_\mu ^{\prime })\,d\sigma . \end{equation*}$$

We use the identity

$$\begin{equation*} \frac{1}{2\pi }\int _0^{2\pi } e^{i(n-m)\sigma }\,d\sigma =\delta _{n,m}, \end{equation*}$$

for the Kronecker delta to obtain an expression for the Hamiltonian function in terms of the oscillators:

$$\begin{equation*} H=\frac{1}{2}\sum _{n\in \mathbb {Z}} (\alpha _{-n}\alpha _n+\beta _{-n}\beta _n), \end{equation*}$$

where we have set $\alpha ^\mu _0=\beta _0^\mu =p^\mu /(2\pi T)$.

The standard Heisenberg–Poisson–Dirac bracket is given by the following expression in the phase variables:

$$\begin{equation*} \begin{aligned} \lbrace x^\mu (\tau ,\sigma ),\dot{x}^\nu (\tau ,\sigma ^{\prime })\rbrace &=\frac{1}{T}\eta ^{\mu ,\nu }\,\delta (\sigma -\sigma ^{\prime }), \\ \lbrace x^\mu (\tau ,\sigma ),x^\nu (\tau ,\sigma ^{\prime })\rbrace &=\lbrace \dot{x}^\mu (\tau ,\sigma ), \dot{x}^\nu (\tau ,\sigma ^{\prime })\rbrace =0. \end{aligned} \end{equation*}$$

Here it is convenient to change to the light variables on $C$, setting the velocity of light equal to 1. These variables are given by $\zeta _\pm =\sigma \pm \tau$, so that the flat metric is $ds_C^2=d\zeta _+\,d\zeta _-$ in these variables. In the light variables the components of the metric $\eta _{\alpha \beta }$ take the form $\eta _{++}=\eta _{--}=0$ and $\eta _{+-}=\eta _{-+}=1/2$, and thus the differentiation operators have the expression $\partial _\pm =(\partial _\sigma \mp \partial _\tau )/2$.

The Virasoro generators are defined by

$$\begin{equation*} \begin{aligned} L_m &=\frac{1}{2}\int _0^{2\pi }T_{++} e^{im(\tau -\sigma )}\,d\sigma =\frac{1}{2}\sum _{n\in \mathbb {Z}} \alpha _{m-n}\alpha _n, \\ \widetilde{L}_m &=\frac{1}{2}\int _0^{2\pi }T_{--} e^{im(\tau +\sigma )}\,d\sigma =\frac{1}{2}\sum _{n\in \mathbb {Z}} \beta _{m-n}\beta _n, \end{aligned} \end{equation*}$$

where $T_{++}$ and $T_{--}$ are the main diagonal elements of the energy-momentum tensor in the real light variables and $\alpha ^\mu _0=\beta _0^\mu =p^\mu /(2\pi T)$. The commutation relations for the operators $L_n$ with respect to the Poisson bracket are

$$\begin{equation*} \lbrace L_m,L_n\rbrace =i(n-m)L_{n+m},\quad \lbrace \widetilde{L}_m,\widetilde{L}_n\rbrace =i(n-m)\widetilde{L}_{n+m},\quad \lbrace L_m,\widetilde{L}_n\rbrace =0. \end{equation*}$$

The next useful operation is the so-called Wick rotation, when the non-physical time $\tau$ is replaced by the purely imaginary $i\tau$. Then the light variables $\zeta _+$ and $\zeta _-$ are replaced by $z$ and $\bar{z}$, where $z=\sigma +i\tau$ and the differentiation operators are replaced as follows: $(\partial _+,\partial _-)\rightarrow (\partial _z,\partial _{\bar{z}})$. The Polyakov action in the conformal scaling becomes

$$\begin{equation*} S_{\mathrm{P}}=-2T\int _C \frac{dz\wedge d\bar{z}}{2i}\,\eta _{\mu \nu }\, \partial _zx^\mu \,\partial _{\bar{z}}x^\nu , \end{equation*}$$

and the equation of motion transforms into the Laplace equation $\partial ^2_{z\bar{z}}x^\mu =0$. The components of the energy-momentum tensor are written as

$$\begin{equation*} \begin{aligned} T_{zz} &=\frac{1}{2}\eta _{\mu \nu } ((x^\mu )^{\prime }-i\dot{x}^\mu )((x^\nu )^{\prime }-i\dot{x}^\nu ) =T_{00}+2iT_{10}, \\ T_{\bar{z}\bar{z}} &=\frac{1}{2} \eta _{\mu \nu }((x^{\mu })^{\prime }+i\dot{x}^\mu ) ((x^\nu )^{\prime }+i\dot{x}^\nu ) =T_{00}-2iT_{10}, \end{aligned} \end{equation*}$$

and $T_{z\bar{z}}=T_{\bar{z}z}=0$. Thus, the Virasoro constraints reduce to $T_{zz}=T_{\bar{z}\bar{z}}=0$ in these variables.

Since $S_{\mathrm{P}}$ is invariant under the infinitesimal reparametrization $\sigma ^\alpha \rightarrow \tilde{\sigma }^\alpha +\varepsilon ^\alpha (\sigma )$, we have the conservation law $\nabla ^\alpha T_{\alpha \beta }=0$ for the energy-momentum tensor, and in the $z$-variables it takes the form

$$\begin{equation*} \partial _{\bar{z}}T_{zz}+\partial _zT_{\bar{z}z} =\partial _zT_{\bar{z}\bar{z}} +\partial _{\bar{z}}T_{z\bar{z}}=0. \end{equation*}$$

Since $T_{z\bar{z}}=T_{\bar{z}z}=0$, we see that $T_{zz}$ is the holomorphic and $T_{\bar{z}\bar{z}}$ the antiholomorphic component of $T$.

Hence in the complex variables we have

$$\begin{equation*} \begin{aligned} \partial _zx^\mu &=\frac{1}{\sqrt{4\pi T}} \sum _{n\in \mathbb {Z}}\beta _n^\mu e^{-inz}, \\ \partial _{\bar{z}}x^\mu &=\frac{1}{\sqrt{4\pi T}} \sum _{n\in \mathbb {Z}} \alpha _n^\mu e^{-in\bar{z}}. \end{aligned} \end{equation*}$$

Thus, the Wick rotation brings about complex coordinates in which the Virasoro operators $L_n$ are the coefficients of the formal Laurent expansions

$$\begin{equation*} T_{zz}=\sum _{n\in \mathbb {Z}} \frac{L_n}{z^{n+2}}, \end{equation*}$$

and the commutation relations for these operators take the form

$$\begin{equation*} \lbrace L_m,L_n\rbrace =i(n-m)L_{n+m}. \end{equation*}$$

In other words, the operators $\lbrace L_n\rbrace$ are the generators of the Witt algebra.

Let us return to the heuristic idea of the integral (7). Assume that we have an 'incoming' string $\ell _{\mathrm{inc}}$ and an 'outgoing' string $\ell _{\mathrm{out}}$. To construct the world sheet we must replace $C$ by a Riemann surface $\Sigma$ of hyperbolic type with fixed oriented boundary $\ell =\ell _{\mathrm{inc}}\cup \ell _{\mathrm{out}}$. Then the flat (Euclidean or Minkowski) target space must be replaced by a curved space, that is, we must introduce a Riemannian or Lorentz metric $h(x)=h_{\mu \nu }\,dz^\mu \,d\bar{z^\nu }$, respectively, where the $z^\mu$ are local coordinates in $\Sigma$ and $x:\Sigma \rightarrow \mathfrak {M}$. The statistical sum $Z[\ell ]$ of the closed bosonic Polyakov string is defined by a perturbation theory series $Z[\ell ]=\sum _{g\geqslant 0}Z_g[\ell ]$ in the genus $g$ of $\Sigma$, where

$$\begin{equation*} \begin{aligned} Z_g[\ell ] &=\int DxDh\,e^{-S_{\mathrm{P}}(x,h)}, \\ S_{\mathrm{P}} &=-2T\int _\Sigma \, \frac{dz\wedge d\bar{z}}{2i} \sqrt{|h|}h^{\mu \nu }\, \partial _zx^\mu \,\partial _{\bar{z}}x^\nu , \end{aligned} \end{equation*}$$

and thus the integration in $S_{\mathrm{P}}$ is performed over the surface $\Sigma$ and we can use conformal coordinates. The path integral in $Z_g[\ell ]$ is taken over the space of Riemann surfaces $\Sigma$ in $\mathfrak {M}$. For its rigorous definition we replace $h$ by a metric $\hat{h}$ with constant curvature on $\Sigma$. The space of such metrics is isomorphic to the space of complex structures on $\Sigma$, which is parametrized by the moduli space $M(g,m)$ with $m$ corresponding to the number of loops in $\ell$. Then the path integral in $Z_g[\ell ]$ over all the metrics reduces to a finite-dimensional integral over the space $M(g,m)$ of complex structures.

For the canonical line bundle $K$ on $\Sigma$ we define an operator $\bar{\partial }_{-1}:T^{(1,0)}_{\mathbb {C}} \Sigma \rightarrow T^{(1,0)}_{\mathbb {C}}\Sigma \otimes \overline{K}$, where $T^{(1,0)}_{\mathbb {C}}\Sigma$ is the holomorphic tangent bundle. In view of the isomorphism $K^{-1}\rightarrow \overline{K}$, we set

$$\begin{equation*} \bar{\partial }_{-1}:\varphi (z)\, \frac{\partial }{\partial z} \mapsto \frac{1}{\rho ^2(z)} \varphi _{\bar{z}}(z). \end{equation*}$$

Here $\rho ^2\,dz\otimes d\bar{z}$ is a conformal metric on $\Sigma$, that is, a non-vanishing section of the bundle $K\otimes \overline{K}$. The adjoint operator of $\bar{\partial }_{-1}$ with respect to the Weil–Petersson Hermitian product coincides with $\bar{\partial }_{-1}^*$.

Using this notation, we obtain the following formal expression for the statistical sum (see, for instance, [42], [49]):

$$\begin{equation*} Z_g[\ell ]=\int _{CF}\,d\sigma \int _{T(g,m)}d_{\mathrm{WP}}\, J(\hat{h}) \biggl (\frac{\det \Delta _{\hat{h}}}{\int _\Sigma \sqrt{\hat{h}}}\biggr )^{-D/2} (\det \bar{\partial }_{-1}^*\,\bar{\partial }_{-1}), \end{equation*}$$

where $d\sigma$ is the formal measure on the space of conformal factors ($CF$) which arises because of the conformal anomaly, and $d_{\mathrm{WP}}$ is the Weil–Petersson measure

$$\begin{equation*} d_{\mathrm{WP}} =\frac{dt_1\wedge \cdots \wedge dt_{6g-6+3m}}{\det (q_i,q_j)}. \end{equation*}$$

It is invariant under the modular group action, and therefore the integration is over the Riemann moduli space $M_{g,m}$. (Here the quadratic differentials $q_j$ form a basis in $T^*T(g,m)$, and the $t_j$ are the coordinates in this basis.) The quantity $J(\hat{h})$ is called the Faddeev–Popov determinant; it can be written as a functional integral over the fields that are Faddeev–Popov ghosts [28], [42], [49]. The main result of bosonic string theory asserts that, in the critical dimension $D=26$, the inner integral is independent of the conformal factor, so the integration over the space of conformal factors can be dropped and the statistical sum turns into a well-defined integral.

Quantization of the Teichmüller space $T(S)=T(g,n)$ corresponding to a compact Riemann surface $S$ of genus $g$ with $n$ punctures was carried out independently by Chekhov and V. V. Fock [31] and Kashaev [51]. Chekhov and Fock used special, so-called 'shear coordinates' on the tangent bundle of the Teichmüller space, while Kashaev dealt with decorated Teichmüller spaces and used the coordinates previously introduced by Penner (the so-called $\lambda$-length). Guo and Liu [39] established connections between these two approaches, first by generalizing the Kashaev algebra, which is a non-commutative deformation of the algebra of rational functions in the Kashaev coordinates and then by establishing a correspondence between the Chekhov–Fock and the Kashaev coordinates. For details see [39] and [84], vol. I, Part D.

We know that any quasiconformal homeomorphism of a Jordan domain $D\subset \widehat{\mathbb {C}}$ onto a Jordan domain $D^{\prime }\subset \widehat{\mathbb {C}}$ extends continuously to the boundary $\partial D$ as a homeomorphism $\overline{D}\rightarrow \overline{D^{\prime }}$ between the closures. We can ask when the converse result holds, that is, when a given homeomorphism between the boundaries of the domains can be extended to a quasiconformal homeomorphism between the domains themselves.

First we examine the case when both domains coincide with the upper half-plane $H=\lbrace z\in \mathbb {C}:\operatorname{Im}z>0\rbrace$. The boundary value of a quasiconformal homeomorphism $H\rightarrow H$ is an orientation-preserving homeomorphism $\mathbb {R}\rightarrow \mathbb {R}$. We say that a monotonically increasing homeomorphism $f:\mathbb {R}\rightarrow \mathbb {R}$ is $k$-quasisymmetric if

$$\begin{equation} \frac{1}{k} \leqslant \frac{f(x+t)-f(x)}{f(x)-f(x-t)}\leqslant k,\qquad x\in \mathbb {R}, \end{equation} \tag{ 13 }$$

for all $t>0$.

It follows from the definition of quasiconformality that the boundary value of a $K$-quasiconformal homeomorphism $H\rightarrow H$ is a $k$-quasisymmetric homeomorphism $\mathbb {R}\rightarrow \mathbb {R}$.

The converse also holds.

Theorem 2 (Beurling–Ahlfors theorem). Let $f:\mathbb {R}\rightarrow \mathbb {R}$ be a $k$-quasisymmetric homeomorphism. Then there exists a quasiconformal homeomorphism $w:H\rightarrow H$ with boundary value equal to $f$ and maximum dilatation bounded by a number $K(k)$ which can be chosen so that $K(k)\rightarrow 1$ as $k\rightarrow 1$.

We give here the idea of the proof (the full proof can be found in [9]).

Let $w$ be the map on the closure $\overline{H}$ defined by

$$\begin{equation*} w(x+iy) =\frac{1}{2}\int _0^1 [f(x+ty)+f(x-iy)]\,dt +\frac{i}{2}\int _0^1[f(x+ty)-f(x-iy)]\,dt. \end{equation*}$$

Obviously, $w(x)=f(x)$ on the real axis. Setting

$$\begin{equation*} \begin{aligned} \alpha (x,y) &=\int _0^1f(x+ty)\,dt =\frac{1}{y}\int _x^{x+y}f(t)\,dt, \\ \beta (x,y) &=\int _0^1f(x-ty)\,dt =\frac{1}{y}\int _{x-y}^xf(t)\,dt, \end{aligned} \end{equation*}$$

we can write the formula for $w$ as follows:

$$\begin{equation*} w(x+iy)=\frac{\alpha +\beta }{2} +i\frac{\alpha -\beta }{2}. \end{equation*}$$

The functions $\alpha$ and $\beta$ have an obvious geometric interpretation: $\alpha (x,y)$ assigns to $x+iy\in H$ the mean value of $f$ on the interval $[x,x+y]$, and $\beta (x,y)$ assigns the mean value of $f$ on $[x-y,x]$.

The functions $\alpha$ and $\beta$ have continuous partial derivatives with respect to $x$ and $y$, and the corresponding map $w:H\rightarrow H$ has a positive Jacobian (calculated in [9]). This shows that, since the boundary value of $w$, which is equal to $f$, is a monotonically increasing homeomorphism $\mathbb {R}\rightarrow \mathbb {R}$, $w$ defines a homeomorphism $H\rightarrow H$. The condition that $w$ is quasiconformal can be expressed by an inequality relating the values of $f$ at the points $x$, $x+y$, and $x-y$, and ensured by the condition (13).

In the case of the unit disk $\Delta$ the Beurling–Ahlfors quasisymmetry condition (13) can be conveniently formulated in terms of the cross-ratio. Recall that the cross-ratio of four distinct points $z_1$, $z_2$, $z_3$, $z_4$ on the Riemann sphere is the quantity

$$\begin{equation*} \mathrm{CR}(z_1,z_2,z_3,z_4) :=\frac{z_4-z_1}{z_4-z_2}: \frac{z_3-z_1}{z_3-z_2}. \end{equation*}$$

Two cross-ratios coincide, $\mathrm{CR}(z_1,z_2,z_3,z_4) =\mathrm{CR}(w_1,w_2,w_3,w_4)$, if and only if there exists a conformal transformation of the Riemann sphere taking $(z_1,z_2,z_3,z_4)$ to $(w_1,w_2,w_3,w_4)$.

We say that an orientation-preserving homeomorphism $f$ of the unit circle $S^1$ onto itself is quasisymmetric if for some $0<\epsilon <1$ it satisfies the condition

$$\begin{equation} \frac{1}{2}(1-\epsilon ) \leqslant \mathrm{CR} \bigl (f(z_1),f(z_2),f(z_3),f(z_4)\bigr ) \leqslant \frac{1}{2}(1+\epsilon ) \end{equation} \tag{ 14 }$$

for an arbitrary quadruple of distinct points $z_1$, $z_2$, $z_3$, $z_4$ on $S^1$ with cross-ratio $\mathrm{CR}(z_1,z_2,z_3,z_4)=1/2$.

The condition (14) is an analogue of the Beurling–Ahlfors condition (13) for the unit disk $\Delta$. As in the case of the upper half-plane $H$, it ensures that the quasisymmetric homeomorphism $f:S^1\rightarrow S^1$ extends quasiconformally into $\Delta$.

Since each orientation-preserving diffeomorphism of $S^1$ onto itself obviously satisfies (14), it is quasisymmetric, that is, can be extended to a quasiconformal diffeomorphism of $\Delta$.

Since the quasiconformal homeomorphisms of the unit disk $\Delta$ onto itself form a group with respect to composition of maps, the quasisymmetric homeomorphisms of the unit circle $S^1$ also form a group with respect to this operation. Let $\mathrm{QS}(S^1)$ denote this group. As mentioned above, $\mathrm{QS}(S^1)$ contains the subgroup $\mathrm{Diff}_+(S^1)$ of orientation-preserving diffeomorphisms of $S^1$. Thus, we have the chain of embeddings

$$\begin{equation*} \operatorname{M\ddot{o}b}(S^1) \subset \mathrm{Diff}_+(S^1) \subset \mathrm{QS}(S^1) \subset \mathrm{Homeo}_+(S^1), \end{equation*}$$

where $\mathrm{Homeo}_+(S^1)$ is the group of orientation-preserving homeomorphisms of $S^1$, and $\operatorname{M\ddot{o}b}(S^1)$ is the group of restrictions to $S^1$ of linear fractional automorphisms of $\Delta$.

Definition 1. The space

$$\begin{equation*} \mathscr T=\mathrm{QS}(S^1)/\operatorname{M\ddot{o}b}(S^1) \end{equation*}$$

is called the universal Teichmüller space.

As already noted, the notion of the universal Teichmüller space was introduced by Bers [17] in 1968. It is very important for the theory of quasiconformal maps and its applications. Recently its relations to string theory were revealed, and they were considered at length by Pekonen in the survey [85].

We can identify $\mathscr T$ with the subspace of $\mathrm{QS}(S^1)$ consisting of normalized quasisymmetric homeomorphisms of $S^1$ which fix three points on the circle. These points are usually taken to be $\pm 1$ and $-i$.

The universal Teichmüller space $\mathscr T$ contains the space

$$\begin{equation*} \mathscr S=\mathrm{Diff}_+(S^1)/\operatorname{M\ddot{o}b}(S^1) \end{equation*}$$

as a subspace which can be identified with the space of normalized quasisymmetric diffeomorphisms of the circle.

Since quasiconformality can be defined in terms of Beltrami differentials, it is convenient to have an explicit definition of $\mathscr T$ in terms of these differentials.

We denote the space of Beltrami differentials in the unit disk $\Delta$ by

$$\begin{equation*} B(\Delta )\equiv B_{(-1,1)}(\Delta ). \end{equation*}$$

It can be identified with the unit ball in the complex Banach space $L^\infty (\Delta )$.

Let $\mu \in B(\Delta )$ be a Beltrami differential in $\Delta$. We extend it to a Beltrami differential $\hat{\mu }$ in the whole extended complex plane $\widehat{\mathbb {C}}$ with the help of symmetry relative to the circle $S^1$; namely, we set

$$\begin{equation*} \hat{\mu }\biggl (\frac{1}{z}\biggr ) :=\overline{\mu (z)}\,\frac{z^2}{\bar{z}^2}\qquad \text{for } z\in \Delta . \end{equation*}$$

Applying the existence theorem for quasiconformal maps to Beltrami's equation with the extended Beltrami differential $\hat{\mu }$, we find a normalized quasiconformal homeomorphism $w_\mu$ of $\widehat{\mathbb {C}}$ with the complex dilatation $\hat{\mu }$. By the uniqueness theorem the homeomorphism $w_\mu$ must be symmetric relative to $S^1$ and therefore must take $S^1$ to itself. Thus, we can associate the normalized quasisymmetric circle homeomorphism $w_\mu$ with the original Beltrami differential $\mu$:

$$\begin{equation*} B(\Delta )\ni \mu \mapsto w_\mu |_{S^1}\in \mathscr T. \end{equation*}$$

This correspondence is one-to-one modulo the following equivalence relation between Beltrami differentials:

$$\begin{equation*} \mu \sim \nu \Longleftrightarrow w_\mu |_{S^1} \equiv w_\nu |_{S^1}. \end{equation*}$$

Hence, the universal Teichmüller space $\mathscr T$ can be identified with the quotient

$$\begin{equation*} \mathscr T=B(\Delta )/{\sim }. \end{equation*}$$

We can also extend the Beltrami differential $\mu \in B(\Delta )$ to another Beltrami differential $\check{\mu }$ on the extended complex plane $\widehat{\mathbb {C}}$ by setting

$$\begin{equation*} \check{\mu }(z)\equiv 0 \text{ for } z\in \Delta _- :=\widehat{\mathbb {C}} \setminus \overline{\Delta }. \end{equation*}$$

Applying the existence theorem for quasiconformal maps to Beltrami's equation with the extended Beltrami differential $\check{\mu }$, we obtain a quasiconformal homeomorphism $w^\mu$ of $\widehat{\mathbb {C}}$ with the complex dilatation $\check{\mu }$. It can be conveniently normalized by fixing the points $0,1,\infty$. The resulting normalized quasiconformal homeomorphism $w^\mu$ is conformal on the complement $\Delta _-$ of the closed unit disk $\overline{\Delta }$.

By a quasidisk we mean the image of the unit disk $\Delta$ under a quasiconformal map, and a quasicircle is the image of the unit circle $S^1$ under such a map. With these definitions, we have just constructed a map

$$\begin{equation*} B(\Delta )\ni \mu \mapsto \text{ quasidisk } \Delta ^\mu :=w^\mu (\Delta ) \text{ to } \widehat{\mathbb {C}}. \end{equation*}$$

It is one-to-one modulo the following equivalence relation between Beltrami differentials:

$$\begin{equation*} \mu \approx \nu \Longleftrightarrow w^\mu |_{\Delta _-} \equiv w^\nu |_{\Delta _-}. \end{equation*}$$

One can show (see [64] and [65]), that the above equivalence relations between Beltrami differentials coincide, that is,

$$\begin{equation*} \mu \sim \nu \Longleftrightarrow \mu \approx \nu . \end{equation*}$$

In view of this, we obtain two further interpretations of the universal Teichmüller space:

$$\begin{equation} \begin{aligned} \mathscr T &=\lbrace \text{the space of normalized quasidisks in $\widehat{\mathbb {C}}$}\rbrace \\ &=\lbrace \text{the space of normalized quasiconformal maps} \\ &\qquad \text{ $\widehat{\mathbb {C}}\rightarrow \widehat{\mathbb {C}}$ which are conformal in $\Delta _-$}\rbrace . \end{aligned} \end{equation} \tag{ 15 }$$

The connection between two realizations of the universal Teichmüller space, as the space of normalized quasisymmetric homeomorphisms of $S^1$ and the space of normalized quasidisks in $\widehat{\mathbb {C}}$, can also be established directly.

To do this we use the following 'welding' problem, which is also of independent interest. Let $f$ be an orientation-preserving homeomorphism of the circle $S^1$. We look for conformal maps $w_+$ and $w_-$, of the disk $\Delta _+$ and its exterior $\Delta _-$, respectively, such that

$$\begin{equation} f=w_+^{-1}\circ w_-\qquad \text{on } S^1. \end{equation} \tag{ 16 }$$

The pair $(w_+,w_-)$ is normalized if the maps $w_\pm$ fix the points $\pm 1$, $-i$.

Lemma 1. Let $f$ be a normalized quasisymmetric homeomorphism of the circle $S^1$ onto itself. Then the welding problem (16) has a unique normalized solution.

In fact, by the Beurling–Ahlfors theorem, for a given homeomorphism $f$ there exists a normalized quasiconformal homeomorphism $w:\Delta _+\rightarrow \Delta _+$ such that $w|_{S^1}=f$. Let $\mu$ denote its complex dilatation. We extend $\mu$ to $\widehat{\mathbb {C}}$ by setting it equal to zero outside $\Delta _+$, which gives us a Beltrami differential $\check{\mu }$, and we let $\widetilde{w}^\mu$ denote the quasiconformal homeomorphism of $\widehat{\mathbb {C}}$ with complex dilatation $\check{\mu }$ that fixes the points $\pm 1,-i\in S^1$. Then the maps

$$\begin{equation} \begin{aligned} w_+&:=\widetilde{w}^\mu |_{\Delta _+} \circ w_\mu ^{-1}:\Delta _+ \rightarrow \text{ the quasidisk } \widetilde{\Delta }^\mu , \end{aligned} \end{equation} \tag{ 17 }$$

$$\begin{equation} w_-:=\widetilde{w}^\mu |_{\Delta _-}:\Delta _- \rightarrow \text{ the quasidisk } \widetilde{\Delta }_-^\mu \end{equation} \tag{ 18 }$$

are conformal in $\Delta _+$ and $\Delta _-$, respectively ($w_+$ is conformal because the homeomorphisms $\widetilde{w}^\mu$ and $w_\mu$ have the same complex dilatation $\mu$ in $\Delta _+$). Hence, they give a normalized solution of the welding problem (16).

Now we return to the correspondence we are interested in:

$$\begin{equation} \left\lbrace {\begin{array}{*{10}c}\text{normalized quasisymmetric} \\ \text{homeomorphisms of $S^1$}\end{array}} \right\rbrace \leftrightarrow \left\lbrace {\begin{array}{*{10}c}\text{normalized} \\ \text{quasidisks in $\widehat{\mathbb {C}}$}\end{array}} \right\rbrace . \end{equation} \tag{ 19 }$$

If $f$ is a normalized quasisymmetric homeomorphism $S^1\rightarrow S^1$, then it has a unique normalized factorization

$$\begin{equation*} f=w_+^{-1}\circ w_-, \end{equation*}$$

where $w_+=\widetilde{w}^\mu |_\Delta \circ w_\mu ^{-1}$ and $w_-=\widetilde{w}^\mu |_{\Delta _-}$. We assign to it the normalized quasidisk $\Delta ^\mu =w^\mu (\Delta )$.

Conversely, if $\Delta ^\mu$ is a normalized quasidisk corresponding to a quasiconformal map $w^\mu$ with complex dilatation $\mu$, then we look at the maps

$$\begin{equation*} w_+=\widetilde{w}^\mu \circ w_\mu ^{-1}\quad \text{on } \Delta _+\quad \text{and}\quad w_-=\widetilde{w}^\mu \quad \text{on } \Delta _-. \end{equation*}$$

These are conformal maps fixing the points $\pm 1$, $-i$ on $S^1$. We assign to them the quasisymmetric homeomorphism of $S^1$ onto itself defined by

$$\begin{equation*} f=w_+^{-1}\circ w_-\quad \text{on } S^1. \end{equation*}$$

This is the required normalized quasisymmetric homeomorphism $S^1\rightarrow S^1$.

In this subsection we present the main properties of the universal Teichmüller space. A more detailed presentation can be found in the monographs [66] and [71] by Lehto and Nag.

We start by describing the topological properties of $\mathscr T$ with respect to the topology defined by the Teichmüller metric (their proofs can be found in [66]):

• 1)  
  $\mathscr T$ is path-connected;
• 2)  
  $\mathscr T$ is complete, that is, any Cauchy sequence in $\mathscr T$ is convergent;
• 3)  
  $\mathscr T$ is contractible;
• 4)  
  $\mathscr T$ is not a topological group, that is, the operation of taking the composite of normalized quasisymmetric homeomorphisms $S^1\rightarrow S^1$ is not continuous in the Teichmüller metric.

To define a complex structure in $\mathscr T$ we embed this space in the space of holomorphic quadratic differentials in the disk. Let $[\mu ]\in \mathscr T$ correspond to a normalized quasiconformal homeomorphism $w^\mu$. Then $w^\mu$ is conformal in the exterior $\Delta _-$ of the unit disk $\Delta$, and we can look at its Schwarzian derivative

$$\begin{equation*} S[w^\mu |_{\Delta _-}]. \end{equation*}$$

We obtain a function holomorphic in $z\in \Delta _-$ which is independent of the choice of $\mu \in [\mu ]$. Moreover, it transforms like a quadratic differential under conformal changes of variable. The map $[\mu ]\mapsto S[w^\mu |_{\Delta _-}]$ is an embedding, since the equality

$$\begin{equation*} S[w^\mu |_{\Delta _-}] =S[w^\nu |_{\Delta _-}] \end{equation*}$$

implies that $w^\mu |_{\Delta _-}=w^\nu |_{\Delta _-}$, and therefore $\mu \sim \nu$.

Thus, we have constructed an embedding

$$\begin{equation*} \Psi :\mathscr T\rightarrow B_2(\Delta _-) \end{equation*}$$

of the universal Teichmüller space $\mathscr T$ in the space $B_2(\Delta _-)$ of holomorphic quadratic differentials in the disk $\Delta _-$; it is called the Bers embedding. The set $B_2(\Delta _-)$ is a complex Banach space, endowed with the natural hyperbolic norm

$$\begin{equation*} B_2(\Delta _-) =\Bigl \lbrace \psi =\psi (z)\,dz^2:\Vert \psi \Vert _{B_2} :=\sup _{z\in \Delta _-}(1-|z|^2)^2|\psi (z)| <\infty \Bigr \rbrace . \end{equation*}$$

The embedding $\Psi$ is a homeomorphism of the space $\mathscr T$ onto its image in $B_2(\Delta _-)$, which is described by the following theorem.

Theorem 3. The image $\Psi (\mathscr T)$ in $B_2(\Delta _-)$ is a connected open contractible subset of $B_2(\Delta _-)$ which contains an open ball $B(0,2)$ with radius 2 and centre at zero and lies in the closed ball $B(0,6)$.

The corresponding proof can be found in [66].

We introduce the complex structure on $\mathscr T$ which is induced by the complex structure on the complex Banach space $B_2(\Delta _-)$ by means of the Bers embedding.

We could also introduce a complex structure on $\mathscr T$ by means of the natural projection

$$\begin{equation*} B(\Delta )\rightarrow \mathscr T=B(\Delta )/{\sim }. \end{equation*}$$

It turns out that both methods give the same result; more precisely we have the following theorem.

Theorem 4 [71]. The composite map of the natural projection $B(\Delta )\rightarrow \mathscr T$ with the Bers embedding

$$\begin{equation*} F:B(\Delta )\rightarrow B_2(\Delta _-) \end{equation*}$$

is a holomorphic map of complex Banach spaces.

We can give an explicit description of the tangent map $dF$ at the point $\mu \equiv 0$. The map $F$ takes this point to the class $[\mathrm{id}]=[\operatorname{M\ddot{o}b}(S^1)]$ in the representation

$$\begin{equation*} \mathscr T=\mathrm{QS}(S^1)/\operatorname{M\ddot{o}b}(S^1). \end{equation*}$$

Let $\mu \in L^\infty (\Delta )$ be an arbitrary tangent vector in $T_0B(\Delta )$. Then for sufficiently small $t$ the function $t\mu$ belongs to $B(\Delta )$ and therefore defines the corresponding normalized quasiconformal homeomorphism $w^{t\mu }$, which has the representation

$$\begin{equation} w^{t\mu }(z)=z+tw_1(z)+o(t) \end{equation} \tag{ 20 }$$

as $t\rightarrow 0$, where $o(t)\equiv t\varepsilon (z,t)$ and $\varepsilon (z,t)\rightarrow 0$ uniformly over $z$ belonging to a compact subset of $\mathbb {C}$.

The coefficient

$$\begin{equation} w_1(z)\equiv \dot{w}[\mu ](z) \end{equation} \tag{ 21 }$$

is the first variation of the quasiconformal homeomorphism $w^\mu$ with respect to $\mu$.

We substitute $w^{t\mu }$ in Beltrami's equation and differentiate this relation with respect to $t$ at $t=0$. Then we get that

$$\begin{equation*} \bar{\partial }w^{t\mu } =t\mu \,\partial w^{t\mu }, \end{equation*}$$

which shows that

$$\begin{equation} \frac{\partial }{\partial t}\biggr |_{t=0} (\bar{\partial }w^{t\mu }) =\mu (\partial w^{t\mu })|_{t=0}. \end{equation} \tag{ 22 }$$

It follows from (20) that

$$\begin{equation*} \partial w^{t\mu }|_{t=0}=1,\qquad \frac{\partial }{\partial t}\bigg |_{t=0} w^{t\mu }=w_1(z), \end{equation*}$$

so from (22) we obtain a $\bar{\partial }$-equation for $w_1(z)$:

$$\begin{equation} \bar{\partial }w_1=\mu , \end{equation} \tag{ 23 }$$

which holds for almost all $z\in \mathbb {C}$. If $\mu$ has compact support, then the solution of this equation is equal to the Cauchy–Green integral

$$\begin{equation*} -\frac{1}{\pi }\int _{\mathbb {C}} \frac{\mu (\zeta )}{\zeta -z}\,d\xi \,d\eta ,\qquad \zeta =\xi +i\eta , \end{equation*}$$

plus some entire function. One can show (see [9]) that this entire function can only be linear, of the form $A+Bz$. It is easy to find the constants $A$ and $B$ from the normalization conditions

$$\begin{equation*} w^{t\mu }(0,t)\equiv 0,\qquad w^{t\mu }(1,t)\equiv 1, \end{equation*}$$

so that $w_1(0)=w_1(1)=0$. It follows from the last relations that

$$\begin{equation*} A=\frac{1}{\pi }\int _{\mathbb {C}} \frac{\mu (\zeta )}{\zeta }\,d\xi \,d\eta ,\qquad B=\frac{1}{\pi }\int _{\mathbb {C}}\mu (\zeta ) \biggl [\frac{1}{\zeta -1} -\frac{1}{\zeta }\biggr ]\,d\xi \,d\eta , \end{equation*}$$

and therefore

$$\begin{equation*} A+Bz=\frac{1-z}{\pi }\int _{\mathbb {C}} \frac{\mu (\zeta )}{\zeta }\,d\xi \,d\eta +\frac{z}{\pi }\int _{\mathbb {C}} \frac{\mu (\zeta )}{\zeta -1}\,d\xi \,d\eta . \end{equation*}$$

Hence, the required solution of (23) is given by

$$\begin{equation} \begin{aligned} w_1(z)&=-\frac{1}{\pi }\int _{\mathbb {C}} \mu (\zeta ) \biggl [\frac{1}{\zeta -z}+\frac{z-1}{\zeta } -\frac{z}{\zeta -1}\biggr ]\,d\xi \,d\eta \\ &=-\frac{z(z-1)}{\pi }\int _{\mathbb {C}} \frac{\mu (\zeta )\,d\xi \,d\eta }{\zeta (\zeta -1)(\zeta -z)}. \end{aligned} \end{equation} \tag{ 24 }$$

(The formula is deduced under the assumption that $\mu$ has compact support in $\mathbb {C}$, but this condition can be dropped with the help of an argument in [9].)

Using (24), we can prove the following result.

Theorem 5 [70]. The differential of the map

$$\begin{equation*} F:B(\Delta )\rightarrow B_2(\Delta _-) \end{equation*}$$

at $\mu \equiv 0$ is a bounded linear operator $d_0F:L^\infty (\Delta )\rightarrow B_2(\Delta _-)$ which is given by

$$\begin{equation*} d_0F[\mu ](z) =-\frac{6}{\pi }\int _\Delta \frac{\mu (\zeta )}{(\zeta -z)^4}\,d\xi \,d\eta ,\qquad z\in \Delta _-,\quad \zeta =\xi +i\eta \in \Delta . \end{equation*}$$

The operator norm of $d_0F$ is bounded by an explicitly calculated constant.

We give the idea of the proof. Fix a point $z_0\in \Delta _-$. We can show that $w^{t\mu }(z)$ is holomorphic in both $z$ and $t$ (and therefore jointly holomorphic) for sufficiently small $|t|$ and $|z-z_0|$. Hence the function

$$\begin{equation*} \varphi (t,z):=S[w^{t\mu }](z), \end{equation*}$$

which coincides with the image of the function $t\mu$ under the Bers map, is holomorphic in the domain

$$\begin{equation*} \lbrace |t|<\epsilon \rbrace \times \lbrace |z-z_0|<\delta \rbrace \subset \mathbb {C}^2 \end{equation*}$$

for sufficiently small $\epsilon$ and $\delta$. We can calculate its $t$-derivative for $t=0$. For brevity we denote the $t$-derivative by a dot and the $z$-derivative by a prime. Then

$$\begin{equation*} \dot{\varphi }=\biggl [\frac{w^{\prime \prime \prime }}{w^{\prime }} -\frac{3}{2}\frac{(w^{\prime \prime })^2}{(w^{\prime })^2}\biggr ]^{\,\cdot \,} =\frac{(w^{\prime })^3\dot{w}^{\prime \prime \prime }-\dot{w}^{\prime }(w^{\prime })^2w^{\prime \prime \prime } -3\dot{w}^{\prime \prime }(w^{\prime })^2w^{\prime \prime }+6\dot{w}^{\prime }w^{\prime }w^{\prime \prime }}{(w^{\prime })^4}. \end{equation*}$$

For $t=0$ we have $w(z)\equiv z$, and therefore $w^{\prime }\equiv 1$ and $w^{\prime \prime }=w^{\prime \prime \prime }\equiv 0$. Hence,

$$\begin{equation*} \frac{\partial \varphi }{\partial t}\biggr |_{t=0} =\frac{(w^{\prime })^3\dot{w}^{\prime \prime \prime }}{(w^{\prime })^4}=\dot{w}^{\prime \prime \prime }. \end{equation*}$$

We now use the formula (24) for $w_1(z)=\dot{w}[\mu ](z)$:

$$\begin{equation*} \dot{w}(z)=-\frac{z(z-1)}{\pi } \int _\Delta \frac{\mu (\zeta )\,d\xi \,d\eta }{\zeta (\zeta -1)(\zeta -z)},\qquad z\in \mathbb {C}. \end{equation*}$$

Since the integrand in the last formula is uniformly bounded for $z$ belonging to a compact subset of $\Delta _-$, we can differentiate with respect to $z$ under the integral sign, obtaining

$$\begin{equation*} \dot{w}^{\prime \prime \prime }(z)=-\frac{6}{\pi } \int _\Delta \frac{\mu (\zeta )}{(\zeta -z)^4}\,d\xi \,d\eta ,\qquad z\in \Delta _-. \end{equation*}$$

An estimate for the norm of $d_0F$ follows from the estimate

$$\begin{equation*} \int _\Delta \,\frac{|d\xi \,d\eta |}{|\zeta -z|^4} \leqslant \frac{\pi }{\mathrm{dist}(z,S^1)^2}\quad \text{for } z\in \Delta _-. \end{equation*}$$

Now we describe the kernel of $d_0F$. To do this we introduce the subspace $A_2(\Delta )$ of $B_2(\Delta )$ consisting of the $L^1$-integrable holomorphic quadratic differentials in the disk $\Delta$:

$$\begin{equation*} A_2(\Delta ) =\biggl \lbrace \psi =\psi (z)\,dz^2\in B_2(\Delta ) :\int _\Delta |\psi (z)|\,|dx\,dy|<\infty \biggr \rbrace . \end{equation*}$$

There exists a natural pairing between this space and the space $B(\Delta )$ of Beltrami differentials:

$$\begin{equation*} \langle \mu ,\psi \rangle =\int _\Delta \mu \psi . \end{equation*}$$

This is the standard pairing between $(-1,1)$- and $(2,0)$-differentials, when we have an integral of an integrable $(1,1)$-form on the right-hand side.

In terms of this pairing the kernel of $d_0F$ can be described as follows.

Lemma 2 (Teichmüller's lemma; see [9]). The kernel of $d_0F$ is the subspace

$$\begin{equation*} N:=A_2(\Delta )^\perp =\lbrace \mu \in L^\infty (\Delta ):\langle \mu ,\psi \rangle =0 \text{ for all } \psi \in A_2(\Delta )\rbrace . \end{equation*}$$

Using these results, we shall try to introduce a Kähler metric in $\mathscr T$. We use the Ahlfors map

$$\begin{equation*} \Phi :L^\infty (\Delta )\rightarrow B_2(\Delta ), \end{equation*}$$

which associates the integral

$$\begin{equation} \Phi [\mu ](z)\equiv \varphi (z) =\int _\Delta \frac{\overline{\mu (\zeta )}}{(1-z\bar{\zeta })^4}\,d\xi \,d\eta \end{equation} \tag{ 25 }$$

with a function $\mu \in L^\infty (\Delta )$. This map assigns to $\mu$ a holomorphic quadratic differential $\varphi =\varphi (z)\,dz^2$ in the disk $\Delta$. The kernel of $\Phi$ coincides with $N=A_2(\Delta )^\perp$.

We wish to use (25) to define a Hermitian metric on $\mathscr T$. First we define the metric at zero and then extend it throughout $\mathscr T$ using the left action of $\mathrm{QS}(S^1)$ on $\mathscr T$. A natural way to define a Hermitian metric on the tangent space $T_0\mathscr T$ is by setting the Hermitian product of tangent vectors $[\mu ],[\nu ]\in T_0\mathscr T=L^\infty (\Delta )/N$ to be equal to the double integral

$$\begin{equation} (\mu ,\nu ) \equiv \langle \mu ,\Phi [\nu ]\rangle =\int _\Delta \int _\Delta \frac{\mu (z)\overline{\nu (\zeta )}}{(1-z\zeta )^4}\,d\xi \,d\eta \,dx\,dy. \end{equation} \tag{ 26 }$$

However, the metric thereby introduced is only defined on a dense subset of $T_0\mathscr T$. This is because the image $\Phi [\nu ]$ in $B_2(\Delta )$ of a general $\nu \in L^\infty (\Delta )$ is not necessarily integrable (that is, does not necessarily lie in $A_2(\Delta )$), and if not, then the integral in (26) is divergent. In fact, (26) is only well defined for sufficiently smooth $[\mu ]$ and $[\nu ]$ in $T_0\mathscr T$.

Let us state this more precisely. Let $[\mu ]\in L^\infty (\Delta )$ be a tangent vector in $T_0\mathscr T$. We look at the map

$$\begin{equation} d\beta :T_0(B(\Delta )/{\sim }) \rightarrow T_{[\operatorname{id}]} (\mathrm{QS}(S^1)/\operatorname{M\ddot{o}b}(S^1)), \end{equation} \tag{ 27 }$$

which is tangent to the isomorphism

$$\begin{equation*} \beta :B(\Delta )/{\sim }\, \rightarrow \mathrm{QS}(S^1)/\operatorname{M\ddot{o}b}(S^1). \end{equation*}$$

It takes the tangent vector $[\mu ]$ to the vector field of the following form on $S^1$:

$$\begin{equation*} v(\theta )\,\frac{d}{d\theta } =\dot{w}[\mu ](z)\,\frac{d}{dz},\qquad z=e^{i\theta }, \end{equation*}$$

where $\dot{w}[\mu ]$ (see (21)) is the first variation of the quasiconformal homeomorphism $w^\mu$ with respect to $\mu$. We call vector fields on $S^1$ that are $d\beta$-images of elements $[\mu ]\in T_0\mathscr T$ quasisymmetric. We show below that the integral in (26) is convergent if the vectors $\mu$ and $\nu$ correspond to vector fields $\dot{w}[\mu ]$ and $\dot{w}[\nu ]$ in the smoothness class $C^{3/2+\epsilon }$ on $S^1$, where $\epsilon >0$ is arbitrary.

Although (26) brings about only a densely defined quasimetric on $\mathscr T$, its restriction to the classical Teichmüller spaces $T(G)$ and the space $\mathscr S$ of normalized differentials is a well-defined Kähler metric (we discuss this in greater detail in the next subsection).

The question of the existence of a Kähler metric on the universal Teichmüller space has been the subject of many papers. We point out the monograph [104] by Takhtajan and Teo, where such a metric was constructed. However, it induces a topology distinct from the one induced by the Teichmüller metric. (In particular, the universal Teichmüller space falls into uncountably many connected components in this topology.)

Now we return to the correspondence (27) considered above and give an intrinsic description of quasisymmetric vector fields on $S^1$ that are the images of vectors $[\mu ]\in T_0(B(\Delta )/{\sim })$ under the map (27). Since the Beurling–Ahlfors condition is essential for this description, it is convenient to start with the case of the upper half-plane $H$.

Let us consider the Zygmund space $\Lambda (\mathbb {R})$ of continuous functions $f:\mathbb {R}\rightarrow \mathbb {R}$ satisfying the conditions

$$\begin{equation*} f(0)=f(1)=0\quad \text{and}\quad \frac{f(x)}{x^2+1}\rightarrow 0\quad \text{as } x\rightarrow \infty \end{equation*}$$

and also satisfying

$$\begin{equation*} |f(x+t)+f(x-t)-2f(x)| \leqslant C|t|\quad \text{for all } x\in \mathbb {R},\ t>0, \end{equation*}$$

for some constant $C>0$. This is a (non-separable) Banach space with the norm

$$\begin{equation*} \Vert f\Vert _\Lambda :=\sup _{x,t} \biggl |\frac{f(x+t)+f(x-t)-2f(x)}{t}\biggr |. \end{equation*}$$

It was shown in [34] that quasisymmetric vector fields on $\mathbb {R}$ correspond precisely to functions in $\Lambda (\mathbb {R})$.

By using the Cayley transform it is now easy to define an analogue of the space $\Lambda (\mathbb {R})$ for the circle $S^1$.

The classical Teichmüller spaces $T(G)$ with $G\subset \operatorname{M\ddot{o}b}(S^1)$ being a Fuchsian group are embedded in the universal Teichmüller space as complex submanifolds.

We recall that a quasisymmetric homeomorphism $f\in \mathrm{QS}(S^1)$ is said to be $G$-invariant with respect to a Fuchsian group $G$ if

$$\begin{equation*} fGf^{-1}\subset \operatorname{M\ddot{o}b}(S^1). \end{equation*}$$

Let $\mathrm{QS}(S^1,G)$ be the subgroup of $G$-invariant quasisymmetric homeomorphisms in $\mathrm{QS}(S^1)$; then we define the classical Teichmüller space $T(G)$ by

$$\begin{equation*} T(G)=\mathrm{QS}(S^1,G)/\operatorname{M\ddot{o}b}(S^1). \end{equation*}$$

The Teichmüller space $T(G)$ is a complex Banach manifold with complex structure induced by the Bers embedding or the natural projection of the space of $G$-invariant Beltrami differentials $B(\Delta ,G)\rightarrow T(G)=B(\Delta ,G)/{\sim }$. The embedding $B(\Delta ,G)\rightarrow B(\Delta )$ induces an embedding of $T(G)$ in the universal Teichmüller space $\mathscr T$ (which corresponds to the Fuchsian group $\lbrace 1\rbrace$) as a complex submanifold.

Let $S_0$ be a Riemann surface uniformized by the Fuchsian group $G$, so that

$$\begin{equation*} S_0=\Delta /G. \end{equation*}$$

For each class $[\mu ]\in T(G)$ we can construct the new Riemann surface

$$\begin{equation*} S_\mu =\Delta /G_\mu , \end{equation*}$$

where $G_\mu =w_\mu Gw_\mu ^{-1}$. The same surface can be represented as

$$\begin{equation*} S_\mu =\Delta ^\mu /G^\mu , \end{equation*}$$

where $\Delta ^\mu :=w^\mu (\Delta )$ and $G^\mu =w^\mu G(w^\mu )^{-1}$. The surfaces $S_\mu =\Delta ^\mu /G^\mu$ are homeomorphic to each other, and differ only in their complex structures. At the same time, the surface $\Delta _-^\mu /G^\mu$, where $\Delta _-^\mu :=w^\mu (\Delta _-)$, is biholomorphically equivalent to $\Delta _-/G$, because $w^\mu$ is conformal on $\Delta _-$.

This is to say that the space $T(G)$ parametrizes, by the correspondence $[\mu ]\mapsto G_\mu$, the various complex structures on the Riemann surface $S_0=\Delta /G$ that can be obtained from the original structure by quasiconformal deformations.

All the properties of the universal Teichmüller space expounded above carry over also to the classical Teichmüller spaces; we simply add the condition of $G$-invariance throughout.

For example, in the case of the unit disk $\Delta$ the Bers embedding is given by the map

$$\begin{equation*} F:B(\Delta ,G)\rightarrow B_2(\Delta _-,G) \end{equation*}$$

which assigns the holomorphic quadratic differential $S[w^\mu |_{\Delta _-}]$ on $\Delta _-$ to a Beltrami differential $\mu \in B(\Delta ,G)$. By definition the space $B_2(\Delta _-,G)$ consists of $G$-invariant holomorphic quadratic differentials in $\Delta _-$ with finite norm

$$\begin{equation*} \Vert \psi \Vert _2:=\sup _{z\in \Delta _-}(1-|z|^2)^2|\psi (z)|<\infty . \end{equation*}$$

The formula for the differential $d_0F$ has the same form:

$$\begin{equation*} d_0F[\mu ](z) =-\frac{6}{\pi }\int _\Delta \frac{\mu (\zeta )}{(\zeta -z)^4}\,d\xi \,d\eta ,\qquad z\in \Delta _-, \end{equation*}$$

for $\mu \in L^\infty (\Delta ,G)$. The kernel of $d_0F$ coincides with the subspace

$$\begin{equation*} N(G)\equiv A_2(\Delta ,G)^\perp =\lbrace \mu \in L^\infty (\Delta ,G) :\langle \mu ,\psi \rangle =0 \text{ for all } \psi \in A_2(\Delta ,G)\rbrace . \end{equation*}$$

Thus, the tangent space to $T(G)$ at the origin coincides with $L^\infty (\Delta ,G)/N(G)$.

As in the case of the universal Teichmüller space, we have the Ahlfors map $L^\infty (\Delta ,G)/N(G)\rightarrow B_2(\Delta ,G)$ given by

$$\begin{equation*} L^\infty (\Delta ,G)\ni \mu \mapsto \Phi [\mu ](z)=\int _\Delta \frac{\overline{\mu (\zeta )}}{(1-z\bar{\zeta })^4}\,d\xi \,d\eta . \end{equation*}$$

As previously, we can try to use this map to introduce a Kähler metric in the space $T(G)$, by defining it on vectors $[\mu ]$ and $[\nu ]$ in

$$\begin{equation*} T_0T(G)=L^\infty (\Delta ,G)/N(G) \end{equation*}$$

by the formula

$$\begin{equation} (\mu ,\nu )_G=\langle \mu ,\Phi [\nu ]\rangle _G :=\int _{\Delta /G}\int _\Delta \frac{\mu (z)\overline{\nu (\zeta )}}{(1-z\bar{\zeta })^4}\,d\xi \,d\eta \,dx\,dy. \end{equation} \tag{ 28 }$$

In the case of the classical Teichmüller spaces $T(G)$, $B_2(\Delta ,G)$ coincides with the space of integrable holomorphic quadratic differentials $A_2(\Delta ,G)$ (see [71]), and we have the well-defined pairing (28) for $\mu ,\nu \in L^\infty (\Delta ,G)$ which determines a Hermitian metric in $T_0T(G)$, the so-called Weil–Petersson metric. This metric on $T(G)$ was introduced and investigated by Weil [120] and Ahlfors [8]. In particular, the latter showed that it has negative holomorphic sectional curvature. A thorough analysis of the properties of this metric and its geodesics was carried out by Wolpert (see [122], [123], [125]). Zograf and Takhtajan (Takhtadzhyan) established an interesting connection between the metric and Liouville's equation (see [127] and [128]).

What can be said about the image of the classical Teichmüller spaces $T(G)$ in the universal Teichmüller space $\mathscr T$? An interesting result due to Bowen [21] shows that this image does not lie in the regular part of $\mathscr T$.

More precisely, we say that a point in $\mathscr T$ is regular if it corresponds to a smooth normalized quasisymmetric homeomorphism in $\mathrm{QS}(S^1)$, or equivalently, to a quasidisk with smooth boundary. Bowen showed that each point in $T(G)\setminus \lbrace 0\rbrace$ corresponds to a quasidisk with a fractal boundary whose Hausdorff dimension $d_H$ satisfies $1<d_H<2$ and can take any value in this interval. In terms of quasisymmetric homeomorphisms it can be shown that if $f$ is a $G$-invariant quasisymmetric homeomorphism which is $C^1$ at least at one point, then $f\in \operatorname{M\ddot{o}b}(S^1)$.

Now we turn to another kind of subspace of $\mathscr T$, the space

$$\begin{equation} \mathscr S=\mathrm{Diff}_+(S^1)/\operatorname{M\ddot{o}b}(S^1) \subset \mathscr T =\mathrm{QS}(S^1)/\operatorname{M\ddot{o}b}(S^1) \end{equation} \tag{ 29 }$$

of normalized diffeomorphisms. It lies entirely in the regular part of $\mathscr T$, and the embedding (29) induces a complex structure on it. However, there is a more straightforward way to introduce this structure on $\mathscr S$.

First of all, note that it suffices to define such a complex structure at the origin $[\operatorname{id}]\in \mathscr S$ and then extend it throughout $\mathscr S$ by the action of the group $\mathrm{Diff}_+(S^1)$. The resulting complex structure will thereby be $\mathrm{Diff}_+(S^1)$-invariant. The tangent space

$$\begin{equation*} T_{[\operatorname{id}]}\mathscr S =T_{[\operatorname{id}]} (\mathrm{Diff}_+(S^1)/\operatorname{M\ddot{o}b}(S^1)) \end{equation*}$$

can be identified with the quotient of the Lie algebra of $\mathrm{Diff}_+(S^1)$ by its subalgebra $\mathrm{sl}(2,\mathbb {R})$ coinciding with the Lie algebra of the Lie group $\operatorname{M\ddot{o}b}(S^1)$. The Lie algebra of the Lie group $\mathrm{Diff}_+(S^1)$ coincides with the algebra $\mathrm{Vect}(S^1)$ of smooth vector fields on $S^1$, whose elements $v=v(\theta )\,\partial /\partial \theta$ are conveniently expressed as Fourier expansions

$$\begin{equation*} v=\sum _{n\in \mathbb {Z}}v_ne_n \end{equation*}$$

with complex coefficients satisfying the condition $\bar{v}_n=v_{-n}$, where the $e_n$ are the basis vector fields

$$\begin{equation*} e_n=e^{in\theta }\,\frac{\partial }{\partial \theta } =iz^{n+1}\,\frac{\partial }{\partial z},\qquad n\in \mathbb {Z},\quad z=e^{i\theta }. \end{equation*}$$

Then the elements $v\in T_{[\operatorname{id}]}\mathscr S$ are given by series of the form

$$\begin{equation*} v=\sum _{n\ne 0,\pm 1}v_ne_n. \end{equation*}$$

The complex structure $J$ on $T_{[\operatorname{id}]}\mathscr S$ is defined by the formula

$$\begin{equation} Jv=-i\sum _{n=2}^\infty v_ne_n +i\sum _{n=-\infty }^{-1}v_ne_n. \end{equation} \tag{ 30 }$$

The complex structure thus introduced is equivalent to the complex structure on $\mathscr T$ constructed above (see [74]).

The space $\mathscr S$ carries a homogeneous symplectic form $\omega$. It is uniquely defined up to a multiplicative constant, and its values on the basis vector fields $e_n\in T_{[\operatorname{id}]}^{\mathbb {C}}\mathscr S$ are equal to (see [97])

$$\begin{equation*} \omega (e_m,e_n)=\alpha (m^3-m)\,\delta _{m,-n},\qquad m,n\in \mathbb {Z}\setminus \lbrace 0,\pm 1\rbrace ,\quad \alpha \in \mathbb {C}\setminus \lbrace 0\rbrace . \end{equation*}$$

From $\omega$ and the complex structure $J$ we can construct a compatible Riemannian metric $g_{\mathrm{R}}$ which is defined on tangent vectors $u,v\in T_{[\operatorname{id}]}\mathscr S$ by

$$\begin{equation*} g_{\mathrm{R}}(u,v) =a\operatorname{Re} \biggl [\sum _{n=2}^\infty \bar{u}_nv_n(n^3-n)\biggr ],\qquad a\in \mathbb {R}\setminus \lbrace 0\rbrace , \end{equation*}$$

where

$$\begin{equation*} u=\sum _{n\ne 0,\pm 1}u_ne_n,\qquad v=\sum _{n\ne 0,\pm 1}v_ne_n. \end{equation*}$$

The Kähler metric

$$\begin{equation} g(u,v)=a\sum _{n=2}^\infty \bar{u}_nv_n(n^3-n) \end{equation} \tag{ 31 }$$

is the complexification of the Riemannian metric $g_{\mathrm{R}}$ so constructed. We note that the series on the right-hand side of (31) is absolutely convergent if $u$ and $v$ are $C^{3/2+\epsilon }$-vector fields with arbitrary $\epsilon >0$.

The Kähler metric (31) on $\mathscr S$ was constructed by Kirillov and Yur'ev [54] (see also [22]). For suitably chosen $a$ it is equal to the restriction to $\mathscr S$ of the Kähler quasimetric (26). More precisely, the following result holds.

Proposition 7 [74]. Let $\mu ,\nu \in L^\infty (\Delta )$ be vectors corresponding to $C^{3/2+\epsilon }$-smooth vector fields $u$ and $v$ on $S^1$, respectively. Then the value of the metric (31) on these fields is

$$\begin{equation*} g(u,v)=\frac{a}{6\pi ^2}\int _\Delta \int _\Delta \frac{\mu (z)\overline{\nu (\zeta )}}{(1-z\bar{\zeta })^4}\,dx\,dy. \end{equation*}$$

As already mentioned, the image of the embedding $\mathscr S\hookrightarrow \mathscr T$ lies in the regular part of $\mathscr T$. On the other hand, the images of the embeddings of the classical Teichmüller spaces $T(G)\setminus \lbrace 0\rbrace$ lie in the irregular part of $\mathscr T$. Thus, these manifolds intersect only at the origin $[\operatorname{id}]\in \mathscr T$.

The reader can find a detailed description of the properties of the space $\mathscr S$ of normalized diffeomorphisms in [97] and in papers cited there.

Here we construct a Grassmannian realization of the universal Teichmüller space given by an embedding of $\mathscr T$ in the Hilbert–Schmidt Grassmannian of infinite-dimensional subspaces of the Sobolev Hilbert space of half-differentiable functions on the circle. This embedding is a consequence of the Nag–Sullivan theorem, a key result in quantization of the universal Teichmüller space. This theorem was proved in [73], where the reader can also find other facts relating to the embedding of the universal Teichmüller space in the infinite-dimensional Grassmannian (see also [90] and [97]).

The Sobolev space of half-differentiable functions is the Hilbert space

$$\begin{equation*} V=H_0^{1/2}(S^1,\mathbb {R}) \end{equation*}$$

of functions $f\in L^2(S^1,\mathbb {R})$ with zero mean on the circle which have a generalized derivative of order $1/2$ in the space $L^2(S^1,\mathbb {R})$. In other words, $V$ consists of functions $f\in L^2(S^1,\mathbb {R})$ whose Fourier series have the form

$$\begin{equation*} f(z)=\sum _{n\ne 0}f_nz^n,\qquad \bar{f}_n=f_{-n},\quad z=e^{i\theta }, \end{equation*}$$

with finite Sobolev norm of order $1/2$:

$$\begin{equation*} \Vert f\Vert _{1/2}^2 =\sum _{n\ne 0}|n|\,|f_n|^2 =2\sum _{n=1}^\infty n|f_n|^2<\infty . \end{equation*}$$

On the space $V$ we consider the antisymmetric symplectic 2-form $\omega :V\times V\rightarrow \mathbb {R}$ defined in terms of the Fourier coefficients of vectors $\xi ,\eta \in V$ by the formula

$$\begin{equation*} \omega (\xi ,\eta ) =-i\sum _{n\ne 0}n\xi _n\eta _{-n} =2\operatorname{Im}\sum _{n=1}^\infty n\xi _n\bar{\eta }_n. \end{equation*}$$

It is well defined in view of the Cauchy–Schwarz inequality

$$\begin{equation*} |\omega (\xi ,\eta )| \leqslant \Vert \xi \Vert _{1/2}\,\Vert \eta \Vert _{1/2}. \end{equation*}$$

The space $V$ also has a complex structure $J^0$ defined in terms of Fourier expansions by

$$\begin{equation} \xi (z)=\sum _{n\ne 0}\xi _nz^n \mapsto (J^0\xi )(z) =-i\sum _{n=1}^\infty \xi _nz^n +i\sum _{n=-\infty }^{-1}\xi _nz^n. \end{equation} \tag{ 32 }$$

This complex structure is compatible with the symplectic form $\omega$ in the following sense: together they define a Riemannian metric on $V$ by the formula $g^0(\xi ,\eta ):=\omega (\xi ,J^0\eta )$, or in terms of the Fourier coefficients,

$$\begin{equation} g^0(\xi ,\eta )=\sum _{n\ne 0}|n|\xi _n\bar{\eta }_n =2\operatorname{Re}\sum _{n=1}^\infty n\xi _n\bar{\eta }_n. \end{equation} \tag{ 33 }$$

In other words, $V$ is a Kähler Hilbert space.

The complexification

$$\begin{equation*} V^{\mathbb {C}}=H_0^{1/2}(S^1,\mathbb {C}) \end{equation*}$$

of $V$ is the complex Hilbert space of functions $f\in L^2(S^1,\mathbb {C})$ with Fourier expansions

$$\begin{equation*} f(z)=\sum _{n\ne 0}f_nz^n,\qquad z=e^{i\theta }, \end{equation*}$$

and finite Sobolev norm:

$$\begin{equation*} \Vert f\Vert _{1/2}^2 =\sum _{n\ne 0}|n|\,|f_n|^2 <\infty . \end{equation*}$$

The Riemannian metric $g^0$ extends to the following Hermitian metric on $V^{\mathbb {C}}$:

$$\begin{equation*} \langle \xi ,\eta \rangle =\sum _{n\ne 0}|n|\xi _n\bar{\eta }_n. \end{equation*}$$

We also take the complex linear extension of $\omega$ and the complex structure $J^0$ to $V^{\mathbb {C}}$. Then $V^{\mathbb {C}}$ can be represented as a direct sum

$$\begin{equation} V^{\mathbb {C}}=W_+\oplus W_-, \end{equation} \tag{ 34 }$$

where $W_\pm$ is the ($\mp i$)-eigenspace of the linear operator $J^0:V^{\mathbb {C}}\rightarrow V^{\mathbb {C}}$. In other words,

$$\begin{equation*} W_+=\biggl \lbrace \xi \in V^{\mathbb {C}}:\xi (z) =\sum _{n=1}^\infty \xi _nz^n\biggr \rbrace \quad \text{and}\quad W_-=\biggl \lbrace \xi \in V^{\mathbb {C}}:\xi (z) =\sum _{n=-\infty }^{-1}\xi _nz^n\biggr \rbrace . \end{equation*}$$

The subspaces $W_\pm$ are isotropic with respect to $\omega$, that is, $\omega (\xi ,\eta )=0$ if $\xi ,\eta \in W_+$ or $\xi ,\eta \in W_-$. The decomposition (34) is an orthogonal direct sum with respect to the Hermitian scalar product $\langle \,\cdot \,,\,\cdot \,\rangle$.

Let $\mathscr D$ denote the Dirichlet space of harmonic functions $h:\Delta \rightarrow \mathbb {R}$ in the disk which are normalized by the condition $h(0)=0$ and have finite energy

$$\begin{equation*} E(h)=\frac{1}{2\pi }\int _\Delta |\operatorname{grad}h(z)|^2\,dx\,dy =\frac{1}{2\pi }\int _\Delta \biggl (\biggl |\frac{\partial h}{\partial x}\biggr |^2 +\biggl |\frac{\partial h}{\partial y}\biggr |^2\biggr )\,dx\,dy <\infty . \end{equation*}$$

The following result is well known (see [73]).

Proposition 8. The Poisson transform

$$\begin{equation*} Pf(z)=\frac{1}{2\pi }\int _0^{2\pi } P(\zeta ,z)f(\zeta )\,d\theta ,\qquad \zeta =e^{i\theta }, \end{equation*}$$

where

$$\begin{equation*} P(\zeta ,z)=\frac{|\zeta |^2-|z|^2}{|\zeta -z|^2} \end{equation*}$$

is the Poisson kernel in $\Delta$, establishes an isometric isomorphism $P:V\rightarrow \mathscr D$ between the Sobolev space $V$ and the Dirichlet space $\mathscr D$ with the norm $\Vert h\Vert _{\mathscr D}^2:=E(h)$.

Let $f$ be an orientation-preserving homeomorphism $S^1\rightarrow S^1$. We associate with it the operator $T_f$ acting on functions $\xi \in V$ by the formula

$$\begin{equation*} (T_f\xi )(z)=\xi (f(z)) -\frac{1}{2\pi }\int _0^{2\pi } \xi (f(e^{i\theta }))\,d\theta ,\qquad z=e^{i\theta }. \end{equation*}$$

Theorem 6 (Nag–Sullivan theorem). The operator $T_f$ acts from $V$ to itself if and only if $f\in \mathrm{QS}(S^1)$. If a quasisymmetric homeomorphism $f$ extends to a $K$-quasiconformal homeomorphism of the disk $\Delta$, then the operator norm of $T_f$ does not exceed $\sqrt{K+K^{-1}}$.

Proof. Sufficiency. Let $f$ be a homeomorphism in $\mathrm{QS}(S^1)$ which extends to a quasiconformal homeomorphism $w$ of $\Delta$. Let $\xi$ be a vector in $V$ and $h=P\xi$ its harmonic extension to $\Delta$. Then the boundary value of the function $g:=h\circ w$ is equal to $\xi \circ f$. We assert that $\xi \circ f\in V$; more precisely, the energy of the harmonic extension of $\xi \circ f$ has the estimate

$$\begin{equation} E(P(\xi \circ f)) \leqslant 2\biggl (\frac{1+k^2}{1-k^2}\biggr )E(Pf), \end{equation} \tag{ 35 }$$

where $k$ is the quasiconformality constant of $w$, which is equal to the norm $\Vert \mu \Vert _\infty$ of the Beltrami differential $\mu$ of $w$. In view of Proposition 8, this will mean that the operator norm of $T_f$ does not exceed

$$\begin{equation*} \Vert T_f\Vert \leqslant \sqrt{2}\sqrt{\frac{1+k^2}{1-k^2}} =\sqrt{K+\frac{1}{K}},\qquad \text{where}\quad K=\frac{1+k}{1-k}. \end{equation*}$$

It is sufficient to prove (35) for the map $g=h\circ w$, because then it will also hold for $P(\xi \circ f)$. (Recall that among all smooth maps with prescribed boundary values the minimum of energy is attained on harmonic maps.) Setting $w=u+iv$, we get that

$$\begin{equation*} \biggl (\frac{\partial g}{\partial x}\biggr )^2 +\biggl (\frac{\partial g}{\partial y}\biggr )^2 \leqslant 2\biggl [\biggl (\frac{\partial h}{\partial u}\biggr )^2 +\biggl (\frac{\partial h}{\partial v}\biggr )^2\biggr ] \bigl (|\partial w|^2+|\bar{\partial }w|^2\bigr )\quad \text{a.e. in } \Delta . \end{equation*}$$

The quasiconformality of $w$ means that

$$\begin{equation*} |\bar{\partial }w|\leqslant k|\partial w|\quad \text{ a.e. in } \Delta , \end{equation*}$$

therefore

$$\begin{equation*} \biggl (\frac{\partial g}{\partial x}\biggr )^2 +\biggl (\frac{\partial g}{\partial y}\biggr )^2 \leqslant 2\biggl [\biggl (\frac{\partial h}{\partial u}\biggr )^2 +\biggl (\frac{\partial h}{\partial v}\biggr )^2\biggr ] \biggl (\frac{1+k^2}{1-k^2}\biggr ) \operatorname{Jac}(w), \end{equation*}$$

where $\operatorname{Jac}(w)=|\partial w|^2 -|\bar{\partial }w|^2$. After a change of variables in the integral for $E(g)$ we obtain the required inequality

$$\begin{equation*} E(g) \leqslant 2\biggl (\frac{1+k^2}{1-k^2}\biggr )E(h). \end{equation*}$$

The Dirichlet integral is known to have the property of conformal invariance. In fact, the above proof demonstrates that it is also quasi-invariant with respect to quasiconformal maps.

Necessity. Using the conformal invariance of the Dirichlet integral, we can reduce the required result to the case of the upper half-plane $H$. Then we take $H^{1/2}(\mathbb {R})$ in place of the Sobolev space $H_0^{1/2}(S^1,\mathbb {R})$. We need the following Douglas formula, which expresses the energy of $f\in H^{1/2}(\mathbb {R})$ in terms of the difference quotient of $f$ (this formula was proved, for instance, in [10]):

$$\begin{equation} E(Pf)=\Vert f\Vert _{1/2}^2 =\frac{1}{4\pi ^2} \int _{\mathbb {R}}\int _{\mathbb {R}} \biggl [\frac{f(x)-f(y)}{x-y}\biggr ]^2\,dx\,dy. \end{equation} \tag{ 36 }$$

Now let $f$ be an orientation-preserving homeomorphism $f:\mathbb {R}\rightarrow \mathbb {R}$ such that

$$\begin{equation*} T_f^{-1}:H^{1/2}(\mathbb {R}) \rightarrow H^{1/2}(\mathbb {R}) \end{equation*}$$

is a bounded operator with norm $M$. We take a cut-off function $\chi _0\in C_0^\infty (\mathbb {R})$, $0\leqslant \chi _0\leqslant 1$, such that $\chi _0\equiv 1$ on $[-1,1]$ and $\chi _0\equiv 0$ outside $[-2,2]$. Fixing $x\in \mathbb {R}$ and $t>0$ we let

$$\begin{equation*} I_1=[x-t,x],\qquad I_2=[x,x+t]. \end{equation*}$$

Consider the shift of $\chi _0$ given by

$$\begin{equation*} \chi _1(s):=\chi _0(as+b), \end{equation*}$$

where we have selected the constants $a$ and $b$ so that $\chi _1\equiv 1$ on $I_1$ and $\chi _1\equiv 0$ on $[x+t,\infty )$.

By assumption, $\chi _1\circ f^{-1}\in H^{1/2}(\mathbb {R})$, and since $T_f^{-1}$ is bounded, it follows that

$$\begin{equation*} \Vert T_f^{-1}\chi _1\Vert =\Vert \chi _1\circ f^{-1}\Vert _{1/2} \leqslant M\Vert \chi _1\Vert _{1/2} =M\Vert \chi _0\Vert _{1/2} \end{equation*}$$

(the last equality follows, for instance, from the Douglas formula). Hence

$$\begin{equation} \begin{aligned} M^2\Vert \chi _0\Vert _{1/2}^2 &\geqslant \Vert \chi _1\circ f^{-1}\Vert _{1/2}^2 =\frac{1}{4\pi ^2} \int _{\mathbb {R}}\int _{\mathbb {R}} \biggl [\frac{\chi _1\circ f^{-1}(r) -\chi _1\circ f^{-1}(s)}{r-s}\biggr ]^2\,dr\,ds \\ &\geqslant \int _{f(x-t)}^{f(x)}\,ds \int _{f(x+t)}^\infty dr\, \frac{1}{(r-s)^2} =\log \biggl [1 +\frac{f(x)-f(x-t)}{f(x+t)-f(x)}\biggr ]. \end{aligned} \end{equation} \tag{ 37 }$$

Consequently,

$$\begin{equation*} \frac{f(x)-f(x-t)}{f(x+t)-f(x)} \geqslant \frac{1}{\exp (M^2\Vert \chi _0\Vert _{1/2}^2)-1} \end{equation*}$$

for all $x\in \mathbb {R}$, $t>0$.

Similarly, taking a shift $\chi _2$ of $\chi _0$ such that $\chi _2\equiv 1$ on $I_2$ and $\chi _2\equiv 0$ on $(-\infty ,x-t]$, we can prove the reverse inequality

$$\begin{equation*} \frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leqslant \exp (M^2\Vert \chi _0\Vert _{1/2}^2)-1. \end{equation*}$$

It follows from the last two inequalities that $f$ satisfies the Ahlfors–Beurling condition (14), that is, $f$ is a quasisymmetric homeomorphism. □

Theorem 7 [73]. The action of the operators $T_f:V\rightarrow V$ with $f\in \mathrm{QS}(S^1)$ on the Sobolev space $V$ preserves the symplectic structure $\omega$:

$$\begin{equation} \omega (T_f\xi ,T_f\eta ) =\omega (\xi ,\eta )\qquad \text{for any }\, \xi ,\eta \in V. \end{equation} \tag{ 38 }$$

Moreover, the complex-linear extension of $T_f$ to the complexified space $V^\mathbb {C}$ preserves the subspaces $W_\pm$ if and only if $f\in \text{M}{\ddot o}\text{b}(S^1)$, and in this case $T_f$ is a unitary operator on $W_\pm$.