We study the Bers isomorphism between the Teichmüller space of the parabolic cyclic group and the universal Teichmüller curve. We prove that this is a group isomorphism and its derivative map gives a remarkable relation between Fourier coefficients of cusp forms and Fourier coefficients of vector fields on the unit circle. We generalize the Takhtajan-Zograf metric to the Teichmüller space of the parabolic cyclic group, and prove that up to a constant, it coincides with the pull back of the Velling-Kirillov metric defined on the universal Teichmüller curve via the Bers isomorphism.
In this paper, we introduce a new asymmetric weak metric on the Teichm{ü}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{ü}ller-Randers metric, is an asymmetric deformation of the Teichm{ü}ller metric, and is obtained by adding to the infinitesimal form of the Teichm{ü}ller metric a differential 1-form. We study basic properties of the Teichm{ü}ller-Randers metric. In the case when the 1-form is exact, any Teichm{ü}ller geodesic between two points is a unique Teichm{ü}ller--Randers geodesic between them. A particularly interes…
The Bers embedding of theTeichmüller space is a homeomorphism into the Banach space of certain holomorphic automorphic forms. For a subspace of the universal Teichmüller space and its corresponding Banach subspace, we consider whether the Bers embedding can project down between their quotient spaces. If this is the case, it is called the quotient Bers embedding. Injectivity of the quotient Bers embedding is the main problem in this paper. Alternatively, we can describe this situation as the universal Teichmüller space having an affine foliated structure induced by this subspace. We give severa…
This survey will appear in Vol. VII of the Hendbook of Teichm{ü}ller theory (European Mathematical Society Publishing House, 2020). It is a commentary on Teichm{ü}ller's paper "Einfache Beispiele zur Wertverteilungslehre", published in 1944, whose English translation appears in that volume. Together with Teichm{ü}ller's paper, we survey the development of value distribution theory, in the period starting from Gauss's work on the Fundamental Theorem of Algebra and ending with the work of Teichm{ü}ller. We mention the foundational work of several mathematicians, including Picard, Laguerre, Poinc…
In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{ü}ller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit sphere at each point in the tangent space to Teichm{ü}ller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichm{ü}ller metric of the torus (which coincides with the the hyperbolic metric). We describe the infinitesimal unit spheres of the metrics in this family.The final version of this paper will appear in Annales Academiæ \ S…
We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{ü}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{ü}lle…
This is a commentary on Teichm{ü}ller's paper Unter-suchungen{ü}ber konforme und quasikonforme Abbildungen (Inves-tigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry , in particular a lemma, known as the Modulsatz, which insures the almost circularity of certain loci defined as complementary components of simply connected regions in the Riemann sphere, and another lemma, which we call the Main Lemma, which insures the circularity near infinity of the image of circles by a qua-siconformal map. The two results find w…
This is a mathematical commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen'' (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teichm{ü}ller wrote on the theory of moduli. It contains the proof of the so-called Teichm{ü}ller existence theorem for a closed surface of genus at least 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichm{ü}ll…
We provide a commentary on Teichm{ü}ller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1940, No.22, 1-197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were rediscovered later on and published without any reference to Teichm{ü}ller.In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they…
The type problem is the problem of deciding, for a simply connected Riemann surface, whether it is conformally equivalent to the complex plane or to the unit dic in the complex plane. We report on Teichm{ü}ller's results on the type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasiconformal map-pings to the type problem) (1937) and Untersuchungen{ü}ber kon-forme und quasikonforme Abbildungen (Investigations on con-formal and quasiconformal mappings) (1938). They concern simply connected Riemann surfaces defined as branched cover…
This is a commentary on Teichm{ü}ller's paper Ein Verschiebungssatz der quasikonformen Abbildung (A displacement theorem of quasiconformal mapping), published in 1944. We explain in detail how Teichm{ü}ller solves the problem of finding the quasiconformal mapping from the unit disc to itself, sending 0 to a strictly negative point on the real line, holding the boundary of the disc pointwise fixed and with the smallest quasiconformal dilatation. We mention also some consequences of this extremal problem and we ask a question.
We outline old and new results concerning the well-known problems in the Teichmüller space theory, i.e., whether these spaces are starlike in the Bers holomorphic embedding and whether any Teichmüller space of dimension greater than 1 is biholomorhically equivalent to bounded convex domain in a complex Banach space.
Nicolas-Auguste Tissot (1824--1897) was a French mathematician and cartographer. He introduced a tool which became known among geographers under the name ``Tissot indicatrix'', and which was widely used during the first half of the twentieth century in cartography. This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude. Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of…
The Bers embebbing realizes the Teichmüller space of a Fuchsian group $G$ as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for $G$. It utilizes the schlicht model of Teichmüller space, where each point is represented by an injective holomorphic function on the disc, and the map is constructed via the Schwarzian differential operator. In this paper we prove that a certain class of differential operators acting on functions of the disc induce holomorphic mappings of Teichmüller spaces, and we also obtain a general formula for the different…
Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with $α$-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local $C^{1+α}$-topology at the base point.
Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $J(f) = J(a_{m_1}, a_{m_2},\dots, a_{m_n}) $ on some classes of univalent functions in the unit disk naturally connected with the canonical class $S$. The given functional $J$ is lifted to the Teichmuller space $\mathbf T_1$ of the punctured disk $\mathbb{D}_{*} = \{0 < |z| < 1\}$ which is biholomorphically equivalent to the Bers…
We construct an Ahlfors-Bers complex analytic model for the Teichmüller space of the universal hyperbolic lamination (also known as Sullivan's Teichmüller space) and the renormalized Weil-Petersson metric on it as an extension of the usual one. In this setting, we prove that Sullivan's Teichmüller space is Kähler isometric biholomorphic to the space of continuous functions from the profinite completion of the fundamental group of a compact Riemann surface of genus greater than or equal to two to the Teichmüller space of this surface; i.e. We find natural Kähler coordinates for the Sullivan's T…
Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus \{\hat{z}_0 \}$. Then we have the composed map $φ\circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $φ: F(S) \rightarrow T(\dot{S})$, where $F(S)$ is the Bers fiber space of $S$, $T(X)$ is the Teichmüller space of $X$ and $U$ is the upper half-plane. The purpose of this paper is to show the map $φ\circ r:T(S)\times U \rightarrow T(\dot{S})$. has a continuous extension to some subset of the boundary $T(S) \times \partial U$.
This is the second paper in a series of investigations of the pluripotential theory on Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type theorem on the boundary behavior of the Poisson integral. We will see a relationship between the pluriharmonic measures and the Patterson-Sullivan measures discussed by Athreya, Bufetov, Eskin and Mirzakhani.
We study isometric maps between Teichmüller spaces and bounded symmetric domains in their intrinsic Kobayashi metric. From a complex analytic perspective, these two important classes of geometric spaces have several features in common but also exhibit many differences. The focus here is on recent results proved by the author; we give a list of open questions at the end. Paper based on plenary talk presented to the 6th Ahlfors-Bers Colloquium at Yale, 23-26 October 2014.
We study the Hilbert manifold structure on $T_{0}(1)$ -- the connected component of the identity of the Hilbert manifold T(1). We characterize points on $T_{0}(1)$ in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators $B_{1}$ and $B_{4}$, associated with the points in $T_{0}(1)$ via conformal welding, are Hilbert-Schmidt. We define a ``universal Liouville action'' -- a real-valued function $\SSS_{1}$ on $T_{0}(1)$, and prove that it is a Kähler potential of the Weil-Petersson metric on $T_{0}(1)$. We also prove that $\SSS_{1}$ is $-\tfrac{1}{12π}$ times the logarithm o…
Similarly to the Bers simultaneous uniformization, the product of the $p$-Weil-Petersson Teichmüller spaces for $p \geq 1$ provides the coordinates for the space of $p$-Weil-Petersson embeddings $γ$ of the real line $\mathbb R$ into the complex plane $\mathbb C$. We prove the biholomorphic correspondence from this space to the $p$-Besov space of $u=\log γ'$ on $\mathbb R$ for $p>1$. From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of $p$-Weil-Petersson curves. In particular, it follows that the correspondenc…
