## Advisory Committee Chair

Lex Oversteegen

## Advisory Committee Members

Alexander Blokh

John Johnstone

Jia Li

Vo Liem

John Mayer

## Document Type

Dissertation

## Date of Award

2017

## Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

## Abstract

Let D be the unit disk in the Riemann sphere C and D ∞ = C\D be its complement. A standard way to uniformize a proper simply connected domain U ⊂ C containing 0 is to consider a Riemann map φ : D → U such that 0 maps to 0. Images in D ∞ of radial line segments are called conformal rays. Brouwer defines metric rays in U using the centers of maximal balls touching ∂U and a function ψ, which we call here the Brouwer map. We use a new definition len of length under which every path has finite length to parameterize metric rays. This produces a radially convex domain which we call Q̃ that preserves more of the structure of the boundary of the domain than the conformal map preserves. We define a function λ̃ : Q̃ → U which maps straight radial line segments of varying length in Q̃ to metric rays in U using the len length parameterization. If ∂U is a locally connected continuum, ∂ Q̃ is a simple closed curve. Otherwise ∂U and ∂ Q̃ are more complicated. For each z in a simply connected domain U containing ∞, the piece of a metric ray through z, from z to ∞, is unique and denoted by R(z). If h : ∂U x [0, 1] → C is an isotopy, starting at the identity, then we denote by U t the unbounded complementary domain of h(∂U x {t}) and by R t (z) the metric ray of U t from z to ∞. We show that for all z ∈ U t there exists a δ and an isotopy H : R t (z) x [t − δ, t + δ] ∩ [0, 1] → C so that H| R t (z)x{t} = id | R t (z) and H R t (z) x {s} = R s (z).

## Recommended Citation

Mann, Ivan Holden, "A Metrically Defined Uniformization Map of Planar Domains" (2017). *All ETDs from UAB*. 2379.

https://digitalcommons.library.uab.edu/etd-collection/2379