We interpolate a new family of Teichmüller spaces $T_{\sharp}^X$ between the universal Teichmüller space $T$ and its little subspace $T_0$, which we call the Teichmüller space of piecewise symmetric homeomorphisms. This is defined by prescribing a subset $X$ of the unit circle. The inclusion relation of $X$ induces a natural inclusion of $T_{\sharp}^X$, and an approximation of $T$ is given by an increasing sequence of $T_{\sharp}^X$. In this paper, we discuss the fundamental properties of $T_{\sharp}^X$ from the viewpoint of the quasiconformal theory of Teichmüller spaces. We also consider the quotient space of $T$ by $T_{\sharp}^X$ as an analog of the asymptotic Teichmüller space.
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…
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 introduce the concept of a new kind of symmetric homeomorphisms on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we endow a complex Banach manifold structure on the space of those generalized symmetric homeomorphisms.
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…
Let $G$ be a finitely generated Kleinian group and let $Δ$ be an invariant collection of components in its region of discontinuity. The Teichmüller space $T(Δ,G)$ supported in $Δ$, is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant under $G$ and supported in $Δ$. In this paper we propose a partial closure of $T(Δ,G)$ by considering certain deformations of the above homeomorphisms. Such a partial closure is denoted by $NT(Δ,G)$ and called the {\it noded Teichmüller space} of $G$ supported in $Δ$. Some concrete examples are discussed.
We consider deformations of a group of circle diffeomorphisms with Hölder continuous derivatives in the framework of quasiconformal Teichmüller theory and show certain rigidity under conjugation by symmetric homeomorphisms of the circle. As an application, we give a condition for such a diffeomorphism group to be conjugate to a Möbius group by a diffeomorphism of the same regularity. The strategy is to find a fixed point of the group which acts isometrically on the integrable Teichmüller space with the Weil-Petersson metric.
In the first part of the paper we describe the complex geometry of the universal Teichmüller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo Möbius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichmüller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert-Schmidt Siegel disc. This qua…
The paper presents some recent results on the Weil-Petersson geometry theory of the universal Teichmüller space, a topic which is important in Teichmüller theory and has wide applications to various areas such as mathematical physics, differential equation and computer vision. \noindent (1) It is shown that a sense-preserving homeomorphism $h$ on the unit circle belongs to the Weil-Petersson class, namely, $h$ can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is squarely integrable in the Poincaré metric if and only if $h$ is absolutely continuous such tha…
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…
We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichmüller theory. An element in this class can be characterized by a property that it can be extended quasiconformally to the upper half-plane so that its complex dilatation induces a vanishing Carleson measure. However, differently from the case on the unit circle, strongly symmetric homeomorphisms on the real line are not preserved under either the composition or the inversion. In this paper, we present the difference and the relation between these two cases. In par…
Quasiconformal homeomorphisms of the unit ball $B^n$ of ${\mathbb R}^n, n \ge 3,$ onto itself with identity boundary values are studied. A spatial analogue of Teichmüller's theorem is proved.
We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In $AdS^3$, if $E$ is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characteriza…
Given a continuous vector field $λ(t, \cdot)$ of Sobolev class $H^{\frac 32}$ on the unit circle $S^1$, the flow maps $η=g(t, \cdot)$ of the differential equation $$ \cases \frac{dη}{dt}=λ(t, η)\\ η(0,ζ)=ζ\endcases $$ are known to be quasisymmetric homeomorphisms. Very recently, Gay-Balmaz-Ratiu [GR] conjectured that the flow curve $g(t, \cdot)$ is in the Weil-Petersson class WP$(S^1)$ and is continuously differentiable with respect to the Hilbert manifold structure of WP$(S^1)$ introduced by Takhtajan-Teo [TT]. The first assertion had already been demonstrated in our previous paper [Sh2]. In …
Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichmüller. Unlike Teichmüller's result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving compl…
For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+ε)$-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space $T(§)$ of the punctured solenoid $§$ under the action of the corresponding Modu…
Teichmüller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given point of the domain to another given point of the domain. For a domain $D \subset {\mathbb R}^n\,,n\ge 2\,,$ we consider the class of all $K$- quasiconformal maps of $D$ onto itself with identity boundary values and Teichmüller's problem in this context. Given a map $f$ of this class and a point $x\in D\,,$ we show that the maximal dilatation of $f$ has a lowe…
We study minimisers of the $p$-conformal energy functionals, \[ \mathsf{E}_p(f):=\int_\ID \IK^p(z,f)\,dz,\quad f|_\IS=f_0|_\IS, \] defined for self mappings $f:\ID\to\ID$ with finite distortion and prescribed boundary values $f_0$. Here \[ \IK(z,f) = \frac{\|Df(z)\|^2}{J(z,f)} = \frac{1+|μ_f(z)|^2}{1-|μ_f(z)|^2}\] is the pointwise distortion functional and $μ_f(z)$ is the Beltrami coefficient of $f$. We show that for quasisymmetric boundary data the limiting regimes $p\to\infty$ recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for …
We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichmüller space $T(\mathbb{H})$ given by $λ\mapsto E^λ|_{S^1}$ is a homeomorphism for the Fréchet topology on $ML_b(\mathbb{H})$ and the Teichmüller topology on $T(\mathbb{H})$, where $E^λ$ is an earthquake with earthquake measure $λ$. A corollary is that earthquakes with discrete earthquake measures are dense in $T(\mathbb{H})$. We also establish infinitesimal versions of the above results.
$L^p$-Extremal Teichmüller mappings between Riemann surfaces are diffeomorphisms
Institute for Advanced Study, Massey University（梅西大学高等研究院）；Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉研究中心）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education, P. R. China（教育部非线性期望前沿科学中心）
