On a relation between the universal Teichmüller space and the Grunsky operator

Being a Strebel point gives a sufficient condition for that the extremal Beltrami coefficient is uniquely determined in a Teichmiiller equivalence class. We consider how Strebel points are characterized. In this paper, we will give a new characterization of Strebel points in a certain subset of the universal Teichmfiller space by a property of the Grunsky operator.     

On Fredholm Eigenvalues of Unbounded Polygons

An important open problem in geometric complex analysis is to establish some algorithms for explicit determination of the basic functionals intrinsically connected with conformal and quasiconformal mappings such as their Teichmüller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This problem has not been solved even for generic quadrilaterals. We provide a restricted solution of the problem for unbounded rectilinear polygons.

     

Grunsky算子的本性模

利用一个推广的Grunsky不等式,借助于单叶函数的拟共形延拓的边界伸缩商,我们给出Grunsky算子的本性模的一些估计.作为推论,我们推出Grunsky算子的紧性准则.     

On Grunsky operator

We discuss the holomorphic dependance and the compactness of the Grunsky operator for a univalent function.     

A Local Expansion Theorem for the Löwner Equation

If W(z) is a power series with complex coefficients which represents an injective function bounded by one in the unit disk and which vanishes at the origin then a Grunsky space $\mathcal{G}(W)$ exists. It is contained contractively in the Dirichlet space for the unit disk. In this paper an admissible family of weighted Dirichlet spaces is used as in the proof of Bieberbach conjecture to construct a Local Grunsky space. An expansion theorem is presented for such a Local Grunsky space. The proof relies on the reproducing kernel function for coefficients of powers of z and Löwner differential equation.     

The Grunsky Norm and Some Coefficient Estimates for Bounded Functions

We estimate the growth order in n of the Taylor coefficientsof bounded univalent functions in the unit disk. Our estimatedepends on the norm of the Grunsky operator. In particular,we improve the known results for coefficients of quasiconformallyextendible univalent functions. 1991 Mathematics Subject Classification30C50, 30C75, 30C55.     

Besov Functions and Tangent Space to the Integrable Teichm\"uller Space

The authors identify the function space which is the tangent space to the integrable Teichmüller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations of this function space are obtained.     

Convergence of earthquake and horocycle paths to the boundary of Teichmüller space

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, which is induced by a quadratic differential whose vertical measured foliation is unique ergodic, converges to the Gardiner-Masur boundary and to the Thurston boundary.     

On the action of Virasoro algebra on the space of univalent functions

We obtain explicit expressions for differential operators defining the action of the Virasoro algebra on the space of univalent functions. We also obtain an explicit Taylor decomposition for Schwarzian derivative and a formula for the Grunsky coefficients.     

A divergent teichmüller geodesic with uniquely ergodic vertical foliation

We construct an example of a quadratic differential whose vertical foliation is uniquely ergodic and such that the Teichmüller geodesic determined by the quadratic differential diverges in the moduli space of Riemann surfaces. This research is partially supported by NSF grant DMS0244472.     

A function model for the Teichmüller space of a closed hyperbolic Riemann surface

We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface.Then we introduce a new metric on the Teichmüller space by using the maximum norm on the function space.We prove that the identity map from the Teichmüller space equipped with the Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Moreover, we prove that the inverse of the identity, i.e., the identity map from the Teichmüller space equipped with this new metric to the Teichmüller space equipped with the Teichmüller metric, is continuous(but not uniformly). Therefore, the topology induced by the new metric is the same as the topology induced by the Teichmüller metric on the Teichmüller space.Finally, we give a remark about the pressure metric on the function model and the Weil-Petersson metric on the Teichmüller space.     

Strebel differentials and Hamilton sequences

Let T(S) be a Teichmüller space of a hyperbolic Riemann surface S, viewed as a set of Teichmüller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any extremal Beltrami differential μ₀ at a given point τ of T(S), there is a Hamilton sequence for μ₀ formed by Strebel differentials in a natural way. Especially, such a kind of Hamilton sequence possesses some special properties. As applications, some results on point shift differentials are given.     

Polygonal quasiconformal maps and grunsky inequalities

There is a deep connection between the Grunsky coefficient inequalities for univalent functions and related extremal quasiconformal maps. In this paper, we develop the technique based on the Grunsky inequalities and apply it to solving a problem concerning polygonal quasiconformal maps. Dedicated to Edger Reich on the occasion of his 75th Birthday.     

An algebra of differential operators and generating functions on the set of univalent functions

With a method close to that of Kirillov [4], we define sequences of vector fields on the set of univalent functions and we construct systems of partial differential equations which have the sequence of the Faber polynomials (F_n) as a solution. Through the Faber polynomials and Grunsky coefficients, we obtain the generating functions for some of the sequences of vector fields.     

Conformal invariants and higher-order schwarz lemmas

We derive a generalization of the Grunsky inequalities using the Dirichlet principle. As a corollary, sharp distortion theorems for bounded univalent functions are proven for invariant differential expressions which are higher-order versions of the Schwarzian derivative. These distortion theorems can be written entirely in terms of conformai invariants depending on the derivatives of the hyperbolic metric, and can be interpreted as ’Schwarz lemmas’. In particular, sharp estimates on distortion of the derivatives of geodesic curvature of a curve under bounded univalent maps are given.     

On Grunsky inequalities and the exact domain of variability of the Grunsky functional

We give a generalized version of the classical Grunsky inequalities. As an application, we study the exact domain of variability of the Grunsky functional for univalent functions with quasiconformal extensions.     

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.     

On local comparison between various metrics on Teichmüller spaces

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). Such 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 them. Each of these spaces is equipped with its own metric, and under some hypotheses, there are inclusions between them. 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 (but 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 on the domain and the classical Teichmüller metric on the range. 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 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.     

Unification of extremal length geometry on Teichmüller space via intersection number

In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We develop the extremal length geometry in the cone which is defined as the inverse image of the compactification via the quotient mapping. The compactification is identified with a subset of the cone by taking an appropriate lift. The cone contains canonically the space of measured foliations in the boundary. We first extend the geometric intersection number on the space of measured foliations to the cone, and observe that the restriction of the intersection number to Teichmüller space is represented by an explicit formula in terms of the Gromov product with respect to the Teichmüller distance. From this observation, we deduce that the Gromov product extends continuously to the compactification. As an application, we obtain an alternative approach to a characterization of the isometry group of Teichmüller space. We also obtain a new realization of Teichmüller space, a hyperboloid model of Teichmüller space with respect to the Teichmüller distance.     

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.