Uniformization by square domains

A Proof of the Uniformization Theorem for Arbitrary Plane Domains

We present a simple constructive proof of the Uniformization Theorem which works for plane domains. The proof is a combination of covering space theory and Koebe's constructive proof of the Riemann mapping theorem, and the resulting algorithm can be used to estimate the Poincar6 metric for the domain.

Simultaneous Uniformization by Lipman Bers

We shall show that any two Riemann surfaces satisfying a certain condition, for instance, any two closed surfaces of the same genus g > l , can be uniformized by one group of fractional linear transformations (Theorem 1). This leads, in conjunction with previous results [2; 3] , to the simultaneous uniformization of all algebraic curves of a given genus (Theorems 2-4). Theorem 5 contains an app...

Uniformization by Classical Schottky Groups

Koebe’s Retrosection Theorem [8] states that every closed Riemann surface can be uniformized by a Schottky group. In [10] Marden showed that non-classical Schottky groups exist, and a first explicit example of a non-classical Schottky group was given by Yamamoto in [14]. Work on Schottky uniformizations of surfaces with certain symmetry has been done by people such as Hidalgo [7]. The natural q...

Uniformization and Anti-uniformization Properties of Ladder Systems

Let S denote a stationary subset of limit ordinals of ω1. A ladder system on S is a sequence {Lα : α ∈ S} such that each Lα is an unbounded subset of α of order type ω. A ladder system is uniformizable if for each sequence 〈fα : α ∈ S〉 of functions fα : Lα → ω there is an F : ω1 → ω such that F Lα =∗ fα for each α ∈ S. I.e., for each α ∈ S, {β ∈ Lα : F (β) 6= fα(β)} is finite. We now formulate ...

Uniformization of Rational Relations