Introduction to Teichmüller theory, old and new, II

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4 Athanase Papadopoulos prescribed stratum. He displays a table comparing the known metric properties of the Teichmüller space of a surface of negative Euler characteristic with corresponding properties of the hyperbolic plane, which, as is well known, is the Teichmüller space of the torus. 1.2 The quasiconformal theory In Chapter 2, Alastair Fletcher and Vladimir Markovic study analytic properties of finite-dimensional as well as infinite-dimensional Teichmüller spaces. They review some classical properties and they present some recent results, in particular concerning biholomorphic maps between Teichmüller spaces. We recall that a Riemann surface is said to be of finite topological type if its fundamental group is finitely generated. It is said of finite analytical type if it is obtained (as a complex space) from a closed Riemann surface by removing a finite set of points. The Teichmüller space T (S) of a Riemann surface S is a Banach manifold which is finite-dimensional if and only if S is of finite analytical type. (Note that T (S) can be infinite-dimensional even if S has finite topological type.) A surface with border has an ideal boundary, which is the union of its ideal boundary curves, and the Teichmüller space of a surface with nonempty border is infinite-dimensional. The most important surface with border is certainly the unit disk D ⊂ C, and its Teichmüller space is called universal Teichmüller space. This space contains all Teichmüller spaces of Riemann surfaces, as we shall recall below. In this chapter, S is a surface of finite or infinite type. The Teichmüller space T (S) of a Riemann surface S is defined as a space of equivalence classes of marked Riemann surfaces (S ′ ,f), with the marking f being a quasiconformal homeomorphism between the base surface S and a Riemann surface S ′ . We recall that for infinite-dimensional Teichmüller spaces, the choice of a base Riemann surface is an essential part of the definition, since homeomorphic Riemann surfaces are not necessarily quasiconformally equivalent. Teichmüller space can also be defined as a space of equivalence classes of Beltrami differentials on a given base Riemann surface. The relation between the two definitions stems from the fact that a quasiconformal mapping from a Riemann surface S to another Riemann surface is the solution of an equation of the form f z = μfz (called a Beltrami equation), with μ a Beltrami differential on S. Fletcher and Markovic also deal with universal Teichmüller space. This is a space of equivalence classes of normalized quasiconformal homeomorphisms of the unit disk D. It is well known that quasiconformal maps of D extend to the boundary ∂D of D. Such quasiconformal maps are normalized so that their extension to the boundary fixes the points 1, −1 and i, and two quasiconformal self-maps of the disk are considered to be equivalent if they induce the same map on ∂D. Like the other Teichmüller spaces, universal Teichmüller space can also be defined as a space of equivalence classes of Beltrami differentials. By lifting quasiconformal homeomorphisms or Beltrami differentials from a surface to the universal cover, the Teichmüller space of any surface
Introduction to Teichmüller theory, old and new, II 5 of hyperbolic type embeds in the universal Teichmüller space, and it is in this sense that the universal Teichmüller space is called “universal”. The complex Banach structure of each Teichmüller space T (S) can be obtained from the so-called Bers embedding of T (S) into the Banach space Q(S) of holomorphic quadratic differentials on the base surface S. In the case where S is of finite analytical type, this embedding provides a natural identification between the cotangent space at a point of T (S) and a Banach space Q of integrable holomorphic quadratic differentials, and the two spaces are finite-dimensional. In the general case, the spaces considered are not necessarily finite-dimensional, and the cotangent space at a point of T (S) is the predual of the Banach space Q, that is, a space whose dual is Q. The predual of Q is called the Bergman space of S. This distinction, which is pointed out by Fletcher and Markovic, is an important feature of the theory of infinite-dimensional Teichmüller spaces. It is well known that the complex-analytic theory of finite-dimensional Teichmüller spaces can be developed using more elementary methods than those that involve the Bers embedding. For instance, for surfaces of finite analytical type,Ahlfors defined the complex structure of Teichmüller space using period matrices obtained by integrating systems of independent holomorphic one-forms over a basis of the homology of the surface. The complex analytic structure on Teichmüller space is then the one that makes the period matrices vary holomorphically. The description of the complex structure in the infinite-dimensional case requires more elaborate techniques. Along the same line, we note some phenomena that occur in infinite-dimensional Teichmüller theory and not in the finite-dimensional one. There is a “mapping class group action” on infinite-dimensional Teichmüller spaces, but, unlike the finite dimensional case, this action is not always discrete. (Here, discreteness means that the orbit of any point under the group action is discrete.) Katsuhiko Matsuzaki studied limit sets and domains of discontinuity for such actions, in the infinite-dimensional case. From the metric-theoretic point of view, Zhong Li and Harumi Tanigawa proved that in each infinite-dimensional Teichmüller space, there are pairs of points that can be connected by infinitely many distinct geodesic segments (for the Teichmüller metric). This contrasts with the finite-dimensional case where the geodesic segment connecting two given points is unique. Li proved non-uniqueness of geodesic segments connecting two points in the universal Teichmüller space, and he showed that there are closed geodesics in any infinite-dimensional Teichmüller space. He also proved that the Teichmüller distance function, in the infinite-dimensional case, is not differentiable at some pairs of points in the complement of the diagonal, in contrast with the finitedimensional case where, by a result of Earle, the Teichmüller distance function is continuously differentiable outside the diagonal. 2 The mention of these differences between the finite- and infinite-dimensional cases will certainly give more importance to the results on isometries and biholomorphic 2 The study of the differentiability of the Teichmüller distance function was initiated by Royden, and it was continued by Earle. More precise results on the differentiability of this function were obtained recently by Mary Rees.
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