On Teichmüller’s proof of Teichmüller’s theorem

A simplified version of Teichmüllers proof (independent of the theory of Beltrami equations with measurable coefficients) of a proposition underlying his continuity argument for the existence part of his theorem on extremal quasiconformal mappings is given. This material is based upon work partially supported by the National Science Foundation under Grant No. NSF MCS-78-27119.     

On Teichmüller’s proof of Teichmüller’s theorem

A simplified version of Teichmüllers proof (independent of the theory of Beltrami equations with measurable coefficients) of a proposition underlying his continuity argument for the existence part of his theorem on extremal quasiconformal mappings is given. This material is based upon work partially supported by the National Science Foundation under Grant No. NSF MCS-78-27119.     

Noded Teichmüller spaces

LetG be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmüller spaceT(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant underG and supported in Δ. In this paper we propose a partial closure ofT(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted byNT(Δ,G) and called thenoded Teichmüller space ofG supported in Δ. Some concrete examples are discussed. Partially supported by Projects Fondecyt 1030252, 1030373, 1040333, Projects UTFSM 12.05.21, 12.05.23 and by grant of the University of Bergen.     

Teichmüller's extremal ring problem

An algorithmic solution to Teichmüller's extremal ring problem is given.     

On the Teichmüller-Wittich-Belinskii Theorem

A survey which highlights this theorem’s impact on value-distribution theory. An application is made to asymptotic values of entire functions, which shows some limitations of the method.     

The closure of Teichmüller space

This research is supported in part by NSF DMS-8700627.     

Harmonic diffeomorphisms and Teichmüller theory

Teichmüller spaces of Riemann surfaces usually are treated by using quasi conformal mappings. We prove the existence of a harmonic diffeomorphism between punctured surfaces. We take this to build up a Teichmüller theory and give a new proof of Teichmüller's theorem for Riemann surfaces of finite type.     

Koebe Problems and Teichmüller Theory

In this paper we study the deformation space of certain Kleinian groups. As a result, we give a new proof of the finite Koebe theorem on Riemann surfaces from a viewpoint of Teichmüller theory. This work is supported by Post–Doctoral Foundation of China     

Biholomorphic isomorphisms between Teichmüller curves

We determine the biholomorphic isomorphisms between Teichmüller curves for torsion free Fuchsian groups. We show that, except in some special cases, a biholomorphic isomorphism between two Teichmüller curves is an allowable mapping; in particular, a biholomorphic automorphism of a Teichmüller curve is induced by an element of the modular group. As a by-product of the proof, we also obtain some topological results on self-mappings of surfaces. Research supported by the National Natural Science Foundation of China.     

Real structures of Teichmüller spaces

The moduli space X^g of compact Riemann surfaces of genus g, g>1, has a canonical antiholomorphic involution. It can easily be defined in terms of complex curves: a point in X^g represented by a curve C is mapped to the point represented by the complex conjugate ¯C of C. In other words, the moduli space has a canonical real structure (cf. Andreotti and Holm [2]). The Teichmüller space has, however, several essentially distinct real structures. The purpose of this note is to describe all real structures of the Teichmüller space T(g,n) of compact Riemann surfaces of genus g punctured at n points.Work supported by the EMIL AALTONEN FOUNDATION     

Riemannian metrics on Teichmüller space

On each compact Riemann surface Σ of genusp≥1, we have the Bergman metric obtained by pulling back the flat metric on its Jacobian via the Albanese map. Taking theL ²-product of holomorphic quadratic differentials w.r.t. this metric induces a Riemannian metric on the Teichmüller spaceT _p that is invariant under the action of the modular group. We investigate geometric properties of this metric as an alternative to the usually employed Weil-Petersson metric. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag.