Some Problems On The Complex Analytic Theory Of Teichmüller Spaces

The main purpose of this dissertation is to deal with the existence of holomophic sections of the projection from Teichmuller curves to Teichmiiller spaces, and of two projections associated with pointed Teichmiiller spaces, and to characterize the biholo-morphic fiber-preserving isomorphisms between two pointed Teichmuller spaces. It is arranged as follows.The first three chapters are preliminary. In Chapter 1, we begin with some back-grounds on Teichmuller spaces, state some open problems on the complex analytic theory of Teichmiiller space and our main results obtained in this dissertation. In Chapter 2, we introduce some basic definitions and results on Teichmuller theory. These include the Teichmuller spaces and modular groups for Riemann surfaces and Fuchsian groups, Bers embedding and complex analytic structure of Teichmuller spaces, Royden-Gardiner theorm on the Kobayashi metric, Bers fiber spaces and Teichmuller curves for Fuchsian groups. In Chapter 3, we introduce three important mappings in Teichmuller theory:Bers-Greenberg isomorphism, puncture-forgetting mapping, and Bers isomorphism.In chapter 4, we shall discuss the existence of holomorphic sections of the natural projectionπ2:V(Γ)→T(Γ) from the Teichmuller curve V(Γ) to the Teichmiiller space T(Γ) whenΓis a torsion free Fuchsian group of infinite type. By means of some recent results on little Teichmiiller spaces and asymptotic Teichmiiller spaces, we obtain a necessary condition for the existence of a holomorphic section ofπ2:V(Γ)→T(Γ). By this necessary condition, we are able to prove that the projectionπ2:V(Γ)→T(Γ) has no holomorphic section whenΓis an elementary torsion free Fuchsian group.Chapter 5 deals with the pointed-Teichmiiller spaces for Fuchsian groups. Let G be a Fuchsian group and g be some pointed-Fuchsian group corresponding to G. We denote by M(G) the set of all Beltrami coefficients for G, and T(g) the corresponding pointed-Teichmiiller space of g. We first discuss the existence of holomorphic sections of the projectionsΦg:M(G)→T(g) andΦ:T(g)→T(G), and show that there exists no holomorphic section in general. Then we discuss the allowable mappings be- tween two pointed Teichmuller spaces, and prove that a biholomorphic fiber-preserving isomorphism between two pointed Teichmuller spaces is in general an allowable map-ping.