Computing Degree of Maps Between Manifolds

We determine the degrees of maps between two given closed （n-1）-connected 2n-mamifolds when n ≡ 1 （mod 8）.     

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.     

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.     

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.     

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.     

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.     

Teichmüller Spaces and Function Spaces

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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.”     

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     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

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 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].     

Lines of Minima and Teichmüller Geodesics

For two measured laminations ν⁺ and ν⁻ that fill up a hyperbolizable surface S and for , let be the unique hyperbolic surface that minimizes the length function e ^t l(ν⁺) + e ^-t l(ν⁻) on Teichmüller space. We characterize the curves that are short in and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface on the Teichmüller 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 ν⁻ around the short curves, we estimate the Teichmüller distance between and . We deduce that this distance can be arbitrarily large, but that if S is a once-punctured torus or four-times-punctured sphere, the distance is bounded independently of t. Received: May 2006, Revision: November 2006, Accepted: February 2007     

Teichmüller曲线的一个同构定理

证明了对于任意一个Fuchs群Γ, 当H/Γ是一个双曲型Riemann曲面时,Teichmüller曲线V(Γ)上有唯一的复流形结构, 使得从Bers纤维空间F(Γ)到V(Γ)的自然投影是全纯的且有局部全纯截面,并推广了如下经典结果: 当H/Γ是紧双曲型Riemann曲面时,V(Γ)只依赖于Γ的型而与Γ的椭圆型元素的阶数无关.     

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.     

The Arnoux–Yoccoz Teichmüller disc

We prove that the Teichmüller disc stabilized by the Arnoux-Yoccoz pseudo-Anosov diffeomorphism contains at least two closed Teichmüller geodesics. This proves that the corresponding flat surface does not have a cyclic Veech group. In addition, we prove that this Teichmüller disc is dense inside the hyperelliptic locus of the connected component (2,2) . The proof uses Ratner’s theorems. Rephrasing our results in terms of quadratic differentials, we show that there exists a holomorphic quadratic differential, on a genus 2 surface, with the two following properties:

On holomorphic sections in Teichmüller spaces

It is proved that, for any elementary torsion free Fuchsian group F, the natural projection from the Teichmiiller curve V（F） to the Teichmiiller space T（F） has no holomorphic section.     

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.