Statistical Hyperbolicity in Teichmüller Space

In this paper we explore the idea that Teichmüller space is hyperbolic “on average.” Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichmüller space. We consider several different measures on Teichmüller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichmüller space, showing that “most triangles are mostly thin.”     

On angles in Teichmüller spaces

We discuss the existence of the angle between two curves in Teichmüller spaces and show that, in any infinite dimensional Teichmüller space, there exist infinitely many geodesic triangles each of which has the same three vertices and satisfies the property that its three sides have the same and arbitrarily given length while its three angles are equal to any given three possibly different numbers from 0 to $\pi $ . This implies that the sum of three angles of a geodesic triangle may be equal to any given number from 0 to $3\pi $ in an infinite dimensional Teichmüller space.     

Teichmüller contraction in the Teichmüller space of a closed set in the sphere

We generalize the principle of Teichmüller contraction and deduce the Hamilton-Krushkaĺ condition for extremal quasiconformal mappings in the Teichmüller space of a closed set in the Riemann sphere.     

Exponential mixing for the Teichmüller flow

We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the $SL(2,\mathbb{R})We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of H?lder observables. A geometric consequence is that the action in the moduli space has a spectral gap.     

An isomorphism theorem for Teichmüller curves

It is proved that for any Fuchsian group Г such that H/Г is a hyperbolic Riemann surface, the Teichmuller curve V(Г) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Г) onto V(Г) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmuller curves is deduced, which generalizes a classical result that the Teichmuller curve V(Г) depends only on the type of Г and not on the orders of the elliptic elements of Г when H/Г is a compact hyperbolic Riemann surface.     

Teichmüller spaces for pointed Fuchsian groups

Let T(G) be the Teichmüller space of a Fuchsian group G and T(G) be the pointed Teichmüller space of a corresponding pointed Fuchsian group G.We will discuss the existence of holomorphic sections of the projection from the space M(G) of Beltrami coefficients for G to T(G) and of that from T(G) to T(G) as well.We will also study the biholomorphic isomorphisms between two pointed Teichmüller spaces.     

Integrable Teichmüller spaces

A new kind of subspaces of the universal Teichmüller space is introduced. Some characterizations of the subspaces are given in terms of univalent functions, Beltrami coefficients and quasisymmetric homeomorphisms of the boundary of the unit disc.     

Stability of quasi-geodesics in Teichmüller space

Let S be a surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2. We show that a Teichmüller quasi-geodesic in the thick part of Teichmüller space for S is contained in a bounded neighborhood of a geodesic if and only if it induces a quasi-geodesic in the curve graph.     

Integrable Teichmüller spaces

A new kind of subspaces of the universal Teichmüller space is introduced. Some characterizations of the subspaces are given in terms of univalent functions, Beltrami coefficients and quasisymmetric homeomorphisms of the boundary of the unit disc.     

Stars at infinity in Teichmüller space

Geometriae Dedicata - We investigate a metric structure on the Thurston boundary of  Teichmüller space. To do this, we develop tools in sup metrics and apply Minsky’s...     

A quantum Teichmüller space

We explicitly describe a noncommutative deformation of the *-algebra of functions on the Teichmüller space of Riemann surfaces with holes that is equivariant with respect to the action of the mapping class group. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 3, pp. 511–528, September, 1999.     

The Teichmüller TQFT volume conjecture for twist knots

The Teichmüller TQFT, defined by Andersen and Kashaev, gives rise to a quantum invariant of triangulated hyperbolic knot complements; it has an associated volume conjecture, where the hyperbolic volume of the knot appears as a certain asymptotic coefficient.In this note, we announce a proof of this volume conjecture for all twist knots up to 14 crossings; along the way we explicitly compute the partition function of the Teichmüller TQFT for the whole infinite family of twist knots.Among other tools, we use an algorithm of Thurston to construct a convenient ideal triangulation of a twist knot complement, as well as the saddle point method for computing limits of complex integrals with parameters.     

Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space II

In this paper, we study the asymptotic behavior of Teichmüller geodesic rays in the Gardiner–Masur compactification. We will observe that any Teichmüller geodesic ray converges in the Gardiner–Masur compactification. Therefore, we get a mapping from the space of projective measured foliations to the Gardiner–Masur boundary by assigning the limits of associated Teichmüller rays. We will show that this mapping is injective but is neither surjective nor continuous. We also discuss the set of points where this mapping is bicontinuous.     

Isometrically embedded polydisks in infinite dimensional Teichmüller spaces

We define isometric holomorphic embeddings of the infinite dimensional polydisk D^∞ in any infinite dimensional Teichmüller space. These embeddings provide simple new proofs that the Teichmüller metric on any infinite dimensional Teichmüller space is non-differentiable and has arbitrarily short simple closed geodesics. They also lead to a complete characterization of the points in Teichmüller space that lie on more than one straight line through the basepoint.     

Closed geodesics and non-convexity in asymptotic Teichmüller spaces

In this paper, the geometric property of asymptotic Teichmüller space is studied. Closed geodesics in any infinite dimensional asymptotic Teichmüller space are constructed, and the non-convexity of spheres in asymptotic Teichmüller space with respect to infinitely many geodesics is proved.     

Teichmüller Spaces and Function Spaces

ＴｅｉｃｈｍüｌｅｒＳｐａｃｅｓａｎｄＦｕｎｃｔｉｏｎＳｐａｃｅｓＧｕｏＨｕｉ（郭辉）（ＳｃｈｏｌｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ，１００８７１）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉＺｈｏｎｇＲ...     

Teichmüller Curves Through Fermat Curves

The billiard in a regular n-gon is known to give rise to a Teichmüller curve. For odd n, we compute the genus of this curve, a number field over which the curve may be defined and branched covering relations between certain pairs of these curves. If n is a power of a prime congruent to 3 or 5 modulo 8, the Teichmüller curve may be defined over the rationals. Received: June 2006, Revision: October 2006, Accepted: November 2006     

Teichmüller spaces and holomorphic motions

The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmüller space of a closed subset of the Reimann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the “Harmonic γ-Lemma” of Bers and Royden. We also study some other applications.     

Teichmüller Extensibility of Circle Homeomorphisms

In this paper a condition is obtained in terms of Dirichlet's integral, for a sense-preserving homeomorphism between the unit circumferences to be prolonged into the interior of disk quasiconformally or as extremal Teichmüller mapping, which sharpens and simplifies the widely known theorems by Teichmüller [ Abh. Preuss. Akad. Wiss. Math. Naturw. Kl. 22 (1939) 1-197], Ahlfors [ J. d'Anal. Math ., 3 (1953/54) 1-98], Hamilton [ Trans. Amer. Math. Soc ., 138 (1969) 399-406], Reich [ Ann. Acad. Sci. Fenn. Ser. A. I. Math . 10 (1985) 469-475], Strebel [ Comment. Math. Helv. , 39 (1964) 77-89], Beurling and Ahlfors [ Acta Math ., 96 (1956) 125-142].