A Uniformization Theorem for Nested Word to Word Transductions

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.

Download Full-text

A discrete uniformization theorem for polyhedral surfaces

Journal of Differential Geometry ◽

10.4310/jdg/1527040872 ◽

2018 ◽

Vol 109 (2) ◽

pp. 223-256 ◽

Cited By ~ 11

Author(s):

Xianfeng David Gu ◽

Feng Luo ◽

Jian Sun ◽

Tianqi Wu

Keyword(s):

Uniformization Theorem ◽

Polyhedral Surfaces

Download Full-text

The uniformization theorem

Conformal Invariants ◽

10.1090/chel/371/10 ◽

2010 ◽

pp. 136-151

Keyword(s):

Uniformization Theorem

Download Full-text

The Robin function and conformal welding – A new proof of the existence

Georgian Mathematical Journal ◽

10.1515/gmj-2017-0057 ◽

2020 ◽

Vol 27 (1) ◽

pp. 43-51

Author(s):

Bodo Dittmar

Keyword(s):

Conformal Mapping ◽

Boundary Value Problem ◽

Harmonic Functions ◽

Mixed Boundary ◽

Mixed Boundary Value Problem ◽

Uniformization Theorem ◽

Robin Function ◽

Connected Case ◽

Simply Connected ◽

Mathematical Physicist

AbstractGreen’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

Download Full-text