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Uniformization problems in the plane: A survey

Dimitrios Ntalampekos

Abstract

In this survey we present the history and recent progress on several fundamental (quasi)conformal uniformization problems in the complex plane. Uniformization refers to the process of mapping a space to a canonical model by means of a well-behaved transformation that preserves the geometry and distorts shapes in a controlled fashion. A central problem in the area is Koebe's conjecture, which remains open after almost 120 years and predicts that each planar domain can be conformally mapped to a circle domain -- that is, a domain whose complementary components are points or closed disks. We trace the history of the conjecture, outline recent developments, and examine the associated uniqueness problem. We also discuss variants, with particular emphasis on the question whether a compact set can be mapped by a quasiconformal self-map of the plane to a Schottky set -- that is, a set in the plane whose complement is the union of disjoint open disks.

Uniformization problems in the plane: A survey

Abstract

In this survey we present the history and recent progress on several fundamental (quasi)conformal uniformization problems in the complex plane. Uniformization refers to the process of mapping a space to a canonical model by means of a well-behaved transformation that preserves the geometry and distorts shapes in a controlled fashion. A central problem in the area is Koebe's conjecture, which remains open after almost 120 years and predicts that each planar domain can be conformally mapped to a circle domain -- that is, a domain whose complementary components are points or closed disks. We trace the history of the conjecture, outline recent developments, and examine the associated uniqueness problem. We also discuss variants, with particular emphasis on the question whether a compact set can be mapped by a quasiconformal self-map of the plane to a Schottky set -- that is, a set in the plane whose complement is the union of disjoint open disks.
Paper Structure (22 sections, 29 theorems, 25 equations, 17 figures, 2 tables)

This paper contains 22 sections, 29 theorems, 25 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conformal uniformization by circle domains
  3. Koebe conjecture
  4. Uniformization by other types of domains
  5. Geometric conditions for uniformization
  6. Uniform domains
  7. Cofat domains
  8. Cospread domains
  9. Gromov hyperbolic domains
  10. Conformal rigidity
  11. Uniformization by exhaustion
  12. Quasiconformal uniformization by Schottky sets
  13. Quasiconformal maps
  14. Quasidisks
  15. Quasiconformal annuli
  16. ...and 7 more sections

Key Result

Theorem 2.2

Every simply connected domain $\Omega\subsetneq \mathbb C$ is conformally equivalent to $\mathbb D$. Moreover, for any two conformal maps $f,g$ from $\Omega$ onto $\mathbb D$ the composition $f\circ g^{-1}$ is a Möbius transformation.

Figures (17)

  • Figure 1: A circle domain. Its boundary can have isolated circles, circles converging to points, isolated points, points converging to circles, Cantor sets, etc.
  • Figure 2: Illustration of Koebe's uniformization theorem for finitely connected domains. Note that a homeomorphism between domains preserves the number of boundary components.
  • Figure 3: The Julia set of the polynomial $z^2+i$ is an example of an $\eta$-spread set because it contains $\eta$-quasitripods in all locations and scales.
  • Figure 4: Left: A geodesic triangle in a Gromov hyperbolic space with the $\delta$-neighborhood of two sides containing the third side. Right: A geodesic triangle in the Euclidean plane, which is not Gromov hyperbolic.
  • Figure 5: Simply connected domains have well-behaved hyperbolic geometry, thanks to the Riemann mapping theorem, but their Euclidean geometry may be highly irregular. On the other hand, the unit disk enjoys both well-behaved Euclidean and hyperbolic geometry. This analogy extends to Gromov hyperbolic and uniform domains by Theorem \ref{['theorem:gromov_uniform']}.
  • ...and 12 more figures

Theorems & Definitions (35)

  • Conjecture 2.1: Koebe conjecture, 1908
  • Theorem 2.2: Riemann mapping theorem, 1851
  • Theorem 2.3: Koebe uniformization theorem, 1920
  • Theorem 2.4: He--Schramm uniformization theorem, 1993
  • Theorem 2.5
  • Theorem 2.6: Brandt:conformalHarrington:conformal
  • Theorem 2.7: HerronKoskela:QEDcircledomainsNtalampekosYounsi:rigidity
  • Theorem 2.8: Schramm:transboundary
  • Theorem 2.9: EsmayliRajala:quasitripod
  • Theorem 2.10: KarafylliaNtalampekos:gromov_hyperbolic
  • ...and 25 more