There are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal Teichmüller space, the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller space, and there are others. In general, these spaces are set-theoretically different. An important question is therefore to understand relations between these spaces. Each of these spaces is equipped with its own metric, and under some hypotheses, there are inclusions between these spaces. In this paper, we obtain local metric comparison results on these inclusions, namely, we show that the inclusions are locally bi-Lipschitz under certain hypotheses. To obtain these results, we use some hyperbolic geometry estimates that give new results also for surfaces of finite type. We recall that in the case of a surface of finite type, all these Teichmüller spaces coincide setwise. In the case of a surface of finite type with no boundary components (and possibly with punctures), we show that the restriction of the identity map to any thick part of Teichmüller space is globally bi-Lipschitz with respect to the length spectrum metric and the classical Teichmüller metric on the domain and on the range respectively. In the case of a surface of finite type with punctures and boundary components, there is a metric on the Teichmüller space which we call the arc metric, whose definition is analogous to the length spectrum metric, but which uses lengths of geodesic arcs instead of lengths of closed geodesics. We show that the restriction of the identity map restricted to any "relative thick" part of Teichmüller space is globally bi-Lipschitz, with respect to any of the three metrics: the length spectrum metric, the Teichmüller metric and the arc metric on the domain and on the range.
A Commentary on Teichm{ü}ller's paper ``Vollst{ä}ndige L{ö}sung einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an extremal problem of the quasiconformal mapping)
我们评论Teichm{ü}ller的文件“Vollst {ä}ndige L{<$}sung einer Extremalaufgabe之quasikonformen Abbildung”“（完整的解决方案的ex-tremal问题的quasiconformal映射），发表于1941年。Teichm{ü}ller在五边形（边界上有五个区别点的圆盘）情形下给出了极值拟共形映射存在性的一个证明.本文的最终版本将出现在第六卷的Teichmüller理论手册的一章。
In this paper, the Teichm{ü}ller spaces of surfaces appear from two points of views: the conformal category and the hyperbolic category. In contrast to the case of surfaces of topologically finite type, the Teichm{ü}ller spaces associated to surfaces of topologically infinite type depend on the choice of a base structure. In the setting of surfaces of infinite type, the Teichm{ü}ller spaces can be endowed with different distance functions such as the length-spectrum distance, the bi-Lipschitz distance, the Fenchel-Nielsen distance, the Teichm{ü}ller distance and there are other distance functi…
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…
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…
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…
The combinatorial structure of the unit tangent spheres and cotangent spheres of Teichm{ü}ller space with Thurston's Finsler metric
Gakushuin University（学习院大学）；Max-Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg（斯特拉斯堡大学法国国家科学研究中心高级数学研究所）
On the early history of moduli and Teichm{ü}ller spaces
我们综述了Riemann模空间和Teichm{ü}ller空间的一些主要贡献。我们的报告具有历史性，但重点是数学思想链。我们从黎曼曲面的介绍开始，以黎曼模空间和Teichm{ü}ller空间的一些基本结构的发现结束。我们指出几个事实，这似乎是未知的许多代数geometers和分析工作的理论。我们感兴趣的时期始于1851年的黎曼，结束于20世纪60年代初，当时Ahlfors和Bers证实了Teichm{üller的结果是正确的。本文是为Linda Keen，Irwin Kra和Rubi Rodriguez编辑的《Lipman Bers，a life in Mathematics》一书撰写的（美国数学学会，2015）。这是专门为纪念李普曼Bers谁首先是一个复杂的分析师，并花了很大一部分他的生命和精力工作的分析结构Teichm{ü}ller空间。他的工作分析，但不可分割的几何和拓扑结构。在这篇综述中，我们强调了这一工作与Riemann，Poincar{é}，Klein，Brouwer，Siegel，Teichm{üller，Weil，Grothendieck等人的工作之间的联系和逻辑依赖。我们解释了这些想法背后的动机。在这样做的时候，我们指出了几个事实，这似乎是未知的许多泰希勒理论家。
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}
这是一个评论Teichm Ller的文件Unter-suchungen Lüber konforme und quasikonforme Abbildungen（Inves-tigations on conformal and quasiconformal mapping）出版于1938年。该文件包含的基本成果在共形几何，特别是一个引理，被称为模，它确保几乎圆形的某些轨迹定义为互补组成部分的简单连通地区的黎曼领域，和另一个引理，我们称之为主要引理，它确保圆形接近无穷大的图像圆的一个准siconformal映射。这两个结果在值分布理论中有着广泛的应用，它们允许有效地利用双连通域和拟共形映射的模。Te-ichm{ü}ller的论文还包含了双连通域共形不变量理论的深入发展。本文的最终版本将出现在\emph{Handbook of Teichm{ü}ller theory}（欧洲数学学会出版社，2020）的第七卷中。
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…
格罗滕迪克的Esquisse d 'un纲领经常被提及，因为它包含了关于dessins d' enfants、Teichm{üller}塔以及绝对伽罗瓦群对这些对象或它们的etale基本群的作用的思想。但这个程序包含了其他几个重要的想法。特别是，由表面拓扑和模空间的黎曼曲面的动机，格罗滕迪克呼吁有重铸的拓扑，以使其适合的对象，半代数和半解析几何，特别是研究的Mumford-Deligne紧化的模空间。本文提出了流形、子流形以及它们之间映射的新概念。我们在本章中回顾这些思想，因为它们与模理论和Te-ichm{ü}ller空间有关。我们还简要地提到格罗滕迪克的想法和早期的理论开发的惠特尼，Lojasiewicz和Hironaka，特别是托姆，并与最近的理论o-最小结构。本文的最终版本将出现在第六卷的Teichmüller理论手册的一章。本卷是献给亚历山大格罗滕迪克的记忆。
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…
