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