Given a compact orientable surface with finitely many punctures $Σ$, let $\Cal S(Σ)$ be the set of isotopy classes of essential unoriented simple closed curves in $Σ$. We determine a complete set of relations for a function from $\Cal S(Σ)$ to $\bold R$ to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on $Σ$. As a conse quence, the Teichmüller space of hyperbolic metrics with geodesic boundary and cusp ends on $Σ$ is reconstructed from an intrinsic $(\bold QP^1, PSL(2, \bold Z))$ structure on $\Cal S(Σ)$.
We comment on Teichm{ü}ller 's paper ''Vollst{ä}ndige L{ö}sung einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an ex-tremal problem of the quasiconformal mapping),, published in 1941. In this paper, Teichm{ü}ller gives a proof of the existence of extremal quasiconformal mappings in the case of the pentagon (disc with five distinguished points on the boundary). The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{ü}ller theory.
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…
For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, we exhibit new geodesics for the arc metric on the Teichm{ü}ller space of an arbitrary surface of negative Euler characteristic with nonempty boundary. We also obtain new geodesics for Thurston's metric on Teichm{ü}ller spaces of hyperbolic surfaces without boundary.
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…
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…
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）
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…
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…
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 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.
We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.
In this paper, we obtain the explicit limit value of the Teichmüller distance between two Teichmüller geodesic rays which are determined by Jenkins-Strebel differentials having a common end point on the augmented Teichmüller space. Furthermore, we also obtain a condition under which these two rays are asymptotic. This is similar to a result of Farb and Masur.
We answer a question of Durham, Hagen, and Sisto, proving that a Teichmüller geodesic ray does not necessarily converge to a unique point in the hierarchically hyperbolic space boundary of Teichmüller space. In fact, we prove that the limit set can be almost anything allowed by the topology.
Given a measured geodesic lamination on a hyperbolic surface, grafting the surface along multiples of the lamination defines a path in Teichmuller space, called the grafting ray. We show that every grafting ray, after reparametrization, is a Teichmuller quasi-geodesic and stays in a bounded neighborhood of a Teichmuller geodesic. As part of our approach, we show that grafting rays have controlled dependence on the starting point. That is, for any measured geodesic lamination Lambda, the map of Teichmuller space which is defined by grafting along Lambda is L-Lipschitz with respect to the Teichm…
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 …
Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conjecture.
We provide a combinatorial condition characterizing curves that are short along a Teichmueller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S x R are also short in the corresponding Teichmueller geodesic, and we provide examples demonstrating that the converse is not true.
We review and organize some results describing the behavior of a Teichmüller geodesic and draw several applications: 1) We show that Teichmüller geodesics do not back track. 2) We show that a Teichmüller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpoints are not necessarily in the thick part. 3) We show that if an edge of a Teichmüller geodesic triangle passes through the thick part, then it is close to one of the other egdes.
Let $X_0$ be a complete borderless infinite area hyperbolic surface. We introduce Thurston's boundary to the Teichmüller space $T(X_0)$ of the surface $X_0$ using Liouville (geodesic) currents. Thurston's boundary to $T(X_0)$ is identified with the space $PML_{bdd}(X_0)$ of projective bounded measured laminations on $X_0$ which naturally extends Thurston's result for closed surfaces. Moreover, the quasiconformal mapping class group $MCG_{qc}(X_0)$ acts continuously on the closure $T(X_0)\cup PML_{bdd}(X_0)$.
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curves along the two geodesics are not identical.
Considering the Teichmüller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are stable. Our main tool is to show that one can get a good estimate for the Lipschitz distance by considering the length ratio of finitely many curves.
The Teichmüller space $\mathcal{T}(Σ)$ of a surface $Σ$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(Σ)$. We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.
For two measured laminations $ν^+$ and $ν^-$ that fill up a hyperbolizable surface $S$ and for $t \in (-\infty, \infty)$, let $L_t$ be the unique hyperbolic surface that minimizes the length function $e^t l(ν^+) + e^{-t} l(ν^-)$ on Teichmuller space. We characterize the curves that are short in $L_t$ and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface $G_t$ on the Teichmuller geodesic whose horizontal and vertical foliations are respectively, $e^t ν^+$ and $e^{-t} ν^-$. By deriving additional information about the twists of $ν^+$ and…
Thurston's boundary to the universal Teichmüller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of Teichmüller type if it shrinks vertical foliation of an integrable holomorphic quadratic differential. In a prior work we established that each Teichmüller geodesic ray limits to a multiple (by the reciprocal of the length of the leaves) of vertical foliation of the quadratic differential. Certain non-integrable holomorphic quadratic differential induce geodesic rays and we consider their …
Let $X$ be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmüller space $T(X)$ of the surface $X$ using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the length spectrum of $X$ is a "closure" of projective bounded measured laminations $PML_{bdd} (X)$, and it coincides with $PML_{bdd}(X)$ when $X$ can be decomposed into a countable union of geodesic pairs of pants whose boundary geodesics $\{α_n\}_{n\in\mathbb{N}}$ have lengths pinched…
Let $S$ be a closed, genus $g$ surface. The space of geodesic currents on $S$ encompasses the set of closed curves up to homotopy, as well as Teichmüller space, and many other spaces of structures on $S$. We show that one can define a mapping class group equivariant, length-minimizing projection from the set of filling geodesic currents down to Teichmüller space, and prove some basic properties of this projection to show that it is well-behaved.
In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem in \cite{Amano14}. We also consider the infimum of the asymptotic distance in shifting base points of the rays along the geodesics. We show that the infimum is represented by two quantities. One is the detour metric between the end points of the rays on the Gardiner-Masur boundary of the Teichmüller space, and the other is the Teichmüller distance between the end points of the rays on the augmented Teichmüller space.
Thurston's boundary to the universal Teichmüller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations $PML_{bdd}(\mathbb{H})$ of $\mathbb{H}$. We prove that each Teichmüller geodesic ray in $T(\mathbb{H})$ has a unique limit point in Thurston's boundary to $T(\mathbb{H})$ unlike in the case of closed surfaces.
The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.
We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into the space of geodesic currents, we prove that a horocycle path in Teichmüller space, induced by a quadratic differential whose vertical measured foliation is unique ergodic, converges to the Gardiner-Masur boundary and to the Thurston boundary.
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 …
We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston's stretch maps and prove the following: (1) On the Teichmueller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length o…
We show that if Teichmüller geodesics spend enough time in the thick part of moduli space, they display CAT(-1)-type properties. In particular, they exponentially contract along strongly stable leaves. As an application we prove two closing lemmas.
We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) there are quadratic differentials so that the limit set is a line segment, c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and d) there are quadratic differentials so that the vertical foliation is ergodic and…
Let S be an oriented surface of genus g with m punctures. If 3g-3+m is at least 4 then we construct for every compact subset K of moduli space a closed Teichmueller geodesic not intersecting K.
We study how the length and the twisting parameter of a curve change along a Teichmuller geodesic. We then use our results to provide a formula for the Teichmuller distance between two hyperbolic metrics on a surface, in terms of the combinatorial complexity of curves of bounded lengths in these two metrics.
In the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense of Thurston. They are directed by complete geodesic laminations that are not chain-recurrent, and they have a nice description in terms of Fenchel-Nielsen coordinates. At the basis of the construction are certain maps with controlled Lipschitz constants between right-angled hyperb…
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodes…
Quasi-convexity of energy functions along Teichmüller geodesics
Korea Institute for Advanced Study (KIAS)（韩国高等科学研究院）；Chongqing University of Technology（重庆理工大学）；Chalmers University of Technology, University of Gothenburg（哥德堡大学查尔姆斯理工大学）
We prove that a certain series defines a constant function using Wolpert's formula for the variation of the length of a geodesic along a Fenchel Nielsen twist. Subsequently we determine the value viewing it as function on the the Deligne Mumford compactification and evaluating it at the stable curve at infinity.
We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over gamma is a hyperbolic metric space. As an application, for complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically inf…
We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only if the geodesic is precompact, i.e. its image in the moduli space is contained in a compact set. Some applications are given, e.g. to stability properties of certain quasi-geodesics in Teichmüller space, and to estimates of translation distance for pseudo-Anosov maps.
A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert's formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.
