scholarly journals Some results on Teichmüller spaces of Klein surfaces

1997 ◽  
Vol 39 (1) ◽  
pp. 65-76
Author(s):  
Pablo Arés Gastesi
Keyword(s):  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Topological Type ◽  
Uniformization Theorem ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Simply Connected Domain ◽  
Nonorientable Surfaces ◽  
Simply Connected ◽  
Orientable Surfaces

The deformation theory of nonorientable surfaces deals with the problem of studying parameter spaces for the different dianalytic structures that a surface can have. It is an extension of the classical theory of Teichmüller spaces of Riemann surfaces, and as such, it is quite rich. In this paper we study some basic properties of the Teichmüller spaces of non-orientable surfaces, whose parallels in the orientable situation are well known. More precisely, we prove an uniformization theorem, similar to the case of Riemann surfaces, which shows that a non-orientable compact surface can be represented as the quotient of a simply connected domain of the Riemann sphere, by a discrete group of Möbius and anti-Möbius transformation (mappings whose conjugates are Mobius transformations). This uniformization result allows us to give explicit examples of Teichmüller spaces of non-orientable surfaces, as subsets of deformation spaces of orientable surfaces. We also prove two isomorphism theorems: in the first place, we show that the Teichmüller spaces of surfaces of different topological type are not, in general, equivalent. We then show that, if the topological type is preserved, but the signature changes, then the deformations spaces are isomorphic. These are generalizations of the Patterson and Bers-Greenberg theorems for Teichmüller spaces of Riemann surfaces, respectively.

scholarly journals On the analytic structure of certain infinite dimensional Teicmüller spaces

1996 ◽  
Vol 141 ◽  
pp. 143-156 ◽  
Author(s):  
Takeo Ohsawa
Keyword(s):  
Riemann Surfaces ◽  
Analytic Structure ◽  
Kobayashi Metric ◽  
Teichmüller Spaces ◽  
Infinite Dimensional ◽  
Teichmuller Spaces ◽  
Teichmüller Metric ◽  
Long Time ◽  
Recent Activity ◽  
Insight Into

It is well known since long time that quasiconformally different finite Riemann surfaces give rise to biholomorphically nonequivalent Teichmüller spaces except for a few obvious cases (cf. [R], [E-K]). This is deduced as an application of Royden’s theorem asserting that the Teichmüller metric is equal to the Kobayashi metric. For the case of infinite Riemann surfaces, however, it is still unknown whether or not the corresponding result holds, although it has been shown by F. Gardiner [G] that Royden’s theorem is also valid for the infinite dimensional Teichmüller spaces. On the other hand, recent activity of several mathematicians shows that the infinite dimensional Teichmüller spaces are interesting objects of complex analytic geometry (cf. [Kru], [T], [N], [E-K-K]). Therefore, based on the generalized form of Royden’s theorem, one might well look for further insight into Teichmüller spaces by studying the above mentioned nonequivalence question.

scholarly journals On the generalized Teichmüller spaces and differential equations

1976 ◽  
Vol 64 ◽  
pp. 97-115 ◽  
Author(s):  
Akikazu Kuribayashi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riemann Surfaces ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Abelian Differential ◽  
The Family

It is well known that for the family F of Riemann surfaces {R(z)} defined by the equations y2 = x(x — l)(x — z), zεC — {0,1}, we have one independent abelian differential ω = y−1dx on each R(z) and if we consider z as a parameter on C — {0,1}, the integrals are solutions of the Gauss’s differential equation

scholarly journals On various Teichmüller spaces of a surface of infinite topological type

2012 ◽  
Vol 140 (2) ◽  
pp. 561-574 ◽  
Author(s):  
Daniele Alessandrini ◽  
Lixin Liu ◽  
Athanase Papadopoulos ◽  
Weixu Su
Keyword(s):  
Topological Type ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces

scholarly journals Teichmuller space theory and classical problems of geometric function theory

2021 ◽  
Vol 18 (2) ◽  
pp. 160-178
Author(s):  
Samue Krushkal
Keyword(s):  
Riemann Surfaces ◽  
Function Theory ◽  
Geometric Function Theory ◽  
Space Theory ◽  
New Approach ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Geometric Function ◽  
Coefficient Problems ◽  
New Applications

Recently the author has presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmüller spaces. It involves the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions.

scholarly journals A geometric characterization of Cn and open balls

1979 ◽  
Vol 75 ◽  
pp. 145-150 ◽  
Author(s):  
Kiyoshi Shiga
Keyword(s):  
Riemann Surface ◽  
Complex Plane ◽  
Unit Disc ◽  
Riemann Sphere ◽  
Geometric Characterization ◽  
Biholomorphic Mapping ◽  
Uniformization Theorem ◽  
Higher Dimensional ◽  
Simply Connected

The purpose of this paper is to give a result concerning the problem of geometric characterizations of the Euclidean n-space Cn and bounded domains. It is well known that a simply connected Riemann surface is biholomorphic to one of the Riemann sphere, the complex plane and the unit disc. And there are several results concerning the geometric characterization of these spaces. To show that some simply connected open Riemann surface is biholomorphic to the complex plane or the unit disc, it is sufficient to see that there exist non constant bounded sub-harmonic functions or not. But in the higher dimensional case, there is no uniformization theorem. By this reason to show that some complex manifold is biholomorphic to Cn or an open ball, we must construct a biholomorphic mapping directly.

scholarly journals A note on the variation of Riemann surfaces

1996 ◽  
Vol 142 ◽  
pp. 1-4 ◽  
Author(s):  
Takeo Ohsawa
Keyword(s):  
Riemann Surface ◽  
Complex Plane ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Universal Covering ◽  
Covering Space ◽  
Uniformization Theorem ◽  
Covering Spaces ◽  
Universal Covering Space

Let X be any Riemann surface. By Koebe’s uniformization theorem we know that the universal covering space of X is conformally equivalent to either Riemann sphere, complex plane, or the unit disc in the complex plane. If X is allowed to vary with parameters we may inquire the parameter dependence of the corresponding family of the universal covering spaces.

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