Stars at infinity in Teichmüller space

Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichm{ü}ller space

These are notes on the hyperbolic geometry of surfaces, Teichm{ü}ller spaces and Thurston's metric on these spaces. They are associated with lectures I gave at the Morningside Center of Mathematics of the Chinese Academy of Sciences in March 2019 and at the Chebyshev Laboratory of the Saint Petersburg State University in May 2019. In particular, I survey several results on the behavior of stretch lines, a distinguished class of geodesics for Thurston's metric and I point out several analogies between this metric and Teichm{ü}ller's metric. Several open questions are addressed. The final versio…

关于锐角三角形的Teichmüller空间

On the Teichm{ü}ller space of acute triangles

该研究延续前人工作，通过直接计算给出锐角三角形子空间上类Thurston度量的距离函数与Finsler结构显式表达式，推导其无穷小刚性，并描述极端Lipschitz映射的最大拉伸轨迹。

Derivatives of length functions and shearing coordinates on teichm{ü}ller spaces

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{ü}ller space. For any isotopy class of closed curves $γ$, we compute the first three derivatives of the length function $\ell\_γ:\mathcal{T}(S)\rightarrow\mathbf{R}\_+$ in the shearing coordinates associated to a maximal geodesic lamination $λ$. We show that if $γ$ intersects every leaf of $λ$, then the Hessian of $\ell\_γ$ is positive-definite. We extend this result to length functions of measured laminations. We also provide a method to compute higher derivatives of length functions of geode…

On the early history of moduli and Teichm{ü}ller spaces

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…

A Commentary on Teichm{ü}ller's paper "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the \emph{Handbook of Teichmüller theory}

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…

TeichmUller's work on the type problem (To appear in Vol. VII of the Handbook of Teichm{ü}ller theory, 2020)

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…

A Commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen''

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…

A commentary on Teichmüller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale"

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…

Teichmüller空间上的水平度量

The horocyclic metric on Teichm{ü}ller spaces

本文从共形视角研究Teichmüller空间上的水平度量，将Thurston的理论扩展到任意亏格的Riemann面，并通过复Legendre变换建立Weil-Petersson symplectic形式的对偶关系。

具有Thurston Finsler度量的Teichmüller空间的单位切球和余切球的组合结构

The combinatorial structure of the unit tangent spheres and cotangent spheres of Teichm{ü}ller space with Thurston's Finsler metric

本文研究了负欧拉示性数闭曲面的Teichmüller空间上Thurston度量诱导的切空间和余切空间范数的单位球面的组合结构，给出了切空间单位球面各面维数的公式，证明了该组合结构独立于Teichmüller空间中的基点，并建立了扩展映射类群与单位球面组合自同构群之间的自然同构（亏格2时为满同态），同时得到了余切空间单位球面面的余维数公式以及面暴露和对应投影加权多曲线的充要条件。

Higher spin mapping class groups and strata of Abelian differentials over Teichm{ü}ller space

For $g \ge 5$, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building off of work of the first author (arXiv:1901.05482), we find that the non-hyperelliptic components are classified by an invariant known as an $r$-spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all $r$-spin stabilizer subgroup…

Teichmüller 空间上的地震度量

The earthquake metric on Teichm{ü}ller space

本文首次系统研究 Thurston 在 Teichmüller 空间上引入的非对称 Finsler 地震度量，证明其若干性质，提出 FD-完备化概念，并将其与 Weil–Petersson 完备化及全局最小化问题联系起来。

泰希米勒空间中单位切球的凸结构

Convex structures of the unit tangent spheres in Teichm{ü}ller space

研究泰希米勒空间中单位切球的凸结构，通过分析瑟斯顿度量的芬斯勒无穷小球，得到面、暴露面和极点的特征，包括面与链回归测地线层叠的关系等，为相关拓扑部分提供了新方法并肯定了一个猜想。

A commentary on Teichmüller's paper "Verschiebungssatz der quasikonformen Abbildung" (A displacement theorem of quasiconformal mapping)

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.

The horofunction compactification of the arc metric on Teichmüller space

The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two compactifications.

On Grothendieck's tame topology

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A …

Geometry of the Gromov product : Geometry at infinity of Teichmüller space

This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a "coarsification" of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller …

The Boundary at Infinity of the Curve Complex and the Relative Teichmüller Space

In this paper we study the boundary at infinity of the curve complex $\mathcal{C}(S)$ of a surface $S$ of finite type and the relative Teichmüller space $\mathcal{T}_{el}(S)$ obtained from the Teichmüller space by collapsing each region where a simple closed curve is short to be a set of diameter 1. $\mathcal{C}(S)$ and $\mathcal{T}_{el}(S)$ are quasi-isometric, and Masur-Minsky have shown that $\mathcal{C}(S)$ and $\mathcal{T}_{el}(S)$ are hyperbolic in the sense of Gromov. We show that the boundary at infinity of $\mathcal{C}(S)$ and $\mathcal{T}_{el}(S)$ is the space of topological equivale…

通过测地流实现的Thurston紧致化：非紧有限面积曲面的情形

Thurston's compactification via geodesic currents: The case of non-compact finite area surfaces

本文提出了一种适用于有限面积曲面的变体方法，解决了Bonahon使用测地流构造Teichm{ü}ller空间Thurston紧致化时仅适用于闭曲面的问题。

Gromov双曲空间与Teichmüller空间上随机等距变换平移长度的线性增长

Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces

该研究针对Gromov双曲空间等距群上的随机游走，证明无矩条件下非初等随机游走平移长度至少线性增长，有限一阶矩下增长率等于漂移，还将技术推广到Teichmüller空间，在有限一阶矩下证明谱定理，拓展了已有结果。

Limit points of lines of minima in Thurston's boundary of Teichmueller space

Given two measured laminations mu and nu in a hyperbolic surface which fill up the surface, Kerckhoff [Lines of Minima in Teichmueller space, Duke Math J. 65 (1992) 187-213] defines an associated line of minima along which convex combinations of the length functions of mu and nu are minimised. This is a line in Teichmueller space which can be thought as analogous to the geodesic in hyperbolic space determined by two points at infinity. We show that when mu is uniquely ergodic, this line converges to the projective lamination [mu], but that when mu is rational, the line converges not to [mu], b…

Thurston边界中的无穷远星

Stars at infinity in the Thurston boundary

该研究针对Thurston和Teichmüller度量下Teichmüller空间的Thurston边界，明确了射影测度叶状结构的星及基于星的性质，扩展了Liu-Shi的定理并否定了Karlsson的猜想。

Blaschke轨迹中的协变度量

The covariance metric in the Blaschke locus

本文证明Blaschke轨迹在Teichmüller空间外为有限维光滑流形，研究其协变Riemann结构，并证明由orbifold的Hitchin表示产生的测地线族具有无限长度。

Comparison Between Teichmuller and Lipschitz Metrics

We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichmuller metric as shown by Minsky), and in the thick part, the two metrics are comparable within additive error. However, these metrics are not comparable in general; we construct a sequence of pairs of points in Teichmuller space whose distances approach zero in the Lipschitz metr…

Divergence of geodesics in Teichmuller space and the mapping class group

We show that both Teichmuller space (with the Teichmuller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmuller space, we find that their divergence is at most quadratic. Furthermore, this estimate is shown to be sharp via examples of pairs of rays with exactly quadratic divergence. The same statements are true for geodesic rays in the mapping class group. We explicitly describe efficient paths "near infinity" in both …

Maximal surfaces and the universal Teichmüller space

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…

Factorization Rules in Quantum Teichmüller Theory

We study the representation theory of the quantum Teichmueller space when going to infinity in the classical Teichmueller space. The geometric ingredients are the extension of Thurston's shear coordinates to the augmented Teichmueller space and the study of the Weil-Petersson Poisson structure for this extension. The result is analogous to the factorization rule found in conformal field theory.

Minimal stretch maps between Euclidean triangle?

Given two triangles whose angles are all acute, we find a homeomorphism with the smallest Lipschitz constant between them and we give a formula for the Lipschitz constant of this map. We show that on the set of pairs of acute triangles with fixed area, the function which assigns the logarithm of the smallest Lipschitz constant of Lipschitz maps between them is a symmetric metric. We show that this metric is Finsler, we give a necessary and sufficient condition for a path in this metric space to be geodesic and we determine the isometry group of this metric space. This study is motivated by Thu…

Minimal discs in hyperbolic space bounded by a quasicircle at infinity

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the boundary of a totally geodesic plane. As a by-product we prove that there is a universal constant C independent of the genus such that if the Teichmüller distance between the ends of a quasi-Fuchsian manifold $M$ is at most C, then $M$ is almost-Fuchsian. The main ingredients of the…

Effective mapping class group dynamics II: Geometric intersection numbers

We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the bou…

Teichmuller geometry of moduli space, II: M(S) seen from far away

We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an explicitly described metric. The main ingredient is Minsky's product regions theorem.

Quantum Teichmüller space from quantum plane

We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator arises from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also show that the quantum univer…

Fenchel-Nielsen coordinates for asymptotically conformal deformations

Let $X$ be an infinite hyperbolic surface endowed with an upper bounded geodesic pants decomposition. Alessandrini, Liu, Papadopoulos, Su and Sun \cite{ALPSS}, \cite{ALPS} parametrized the quasiconformal Teichmüller space $T_{qc}(X)$ and the length spectrum Teichmüller space $T_{ls}(X)$ using the Fenchel-Nielsen coordinates. A quasiconformal map $f:X\to Y$ is said to be {\it asymptotically conformal} if its Beltrami coefficient $μ=\bar{\partial}f/\partial f$ converges to zero at infinity. The space of all asymptotically conformal maps up to homotopy and post-composition by conformal maps is ca…

Half-Pipe空间中的平均曲面与无穷小Teichmüller理论

Mean surfaces in Half-Pipe space and infinitesimal Teichmüller theory

该研究建立了Half-Pipe空间中平均曲面与双曲平面上调和拉格朗日向量场的对应关系，证明了无穷小Douady-Earle延拓的相关性质，得到圆周向量场调和拉格朗日延拓的存在唯一性，并刻画了Zygmund类条件的定量界。

Word length statistics for Teichmüller geodesics and singularity of harmonic measure

Given a measure on the Thurston boundary of Teichmueller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a distribution with finite first moment in the word metric. We consider the ratio between the word metric and the relative metric of approximating mapp…

Growth of the Weil-Petersson inradius of moduli space

In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$. Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of t…

通过 Thurston 理论研究函数论、动力学与遍历理论

Function theory, Dynamics and Ergodic theory via Thurston theory

该研究借助 Thurston 理论考察 Teichmüller 空间函数论与映射类群子群动力学，并联系 Sullivan 的双曲等距群离散子群遍历作用理论。

层级双曲空间中的无穷方体化

Cubulating Infinity in Hierarchically Hyperbolic Spaces

该研究证明层级双曲空间中有限内点、射线与边界点的层级包拟中位数拟等距于有界维CAT(0)立方复形，其边界同构，还给出立方模型构造及距离编码应用。

The convex core of quasifuchsian manifolds with particles

We consider quasifuchsian manifolds with "particles", i.e., cone singularities of fixed angle less than $π$ going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particles. We prove that this defines a homeomorphism from the space of quasifuchsian metrics with $n$ particles (of fixed angle) and the product of two copies of the Teichmüller space of a surface with $n$ marked points. This is analoguous to the Bers theorem in the non-singular case. Q…

Flat Metrics, Cubic Differentials and Limits of Projective Holonomies

Labourie and the author independently showed that a convex real projective structure on an oriented surface of genus at least 2 is equivalent to a conformal structure plus a holomorphic cubic differential U. We analyze the behavior of the real-projective structure as the conformal structure is fixed and the cubic differential is scaled to infinity. In particular, we find the asymptotic eigenvalues of the holonomies around smooth geodesic loops with respect to the flat metric given by the 2/3 power of |U|. If the asymptotic holonomies of all loops could be computed, then that would describe a b…

Constant Gaussian curvature foliations and Schläfli formulas of hyperbolic $3$-manifolds

We study the geometry of the foliation by constant Gaussian curvature surfaces $(Σ_k)_k$ of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the Thurston and the Schwarzian parametrizations are the limits of two families of parametrizations of the space of hyperbolic ends, defined by Labourie in 1992 in terms of the geometry of the leaves $Σ_k$. We give a new description of the renormalized volume using the constant curvature foliation. We prove a generalization of McMullen's Kleinian reciprocity theorem, which …

交叉比与超几何群的立方化

Cross ratios and cubulations of hyperbolic groups

本文研究了超几何群的立方化，并展示了其与边界上的不变交叉比的关系，证明了立方化在长度谱上的刚性。