Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized quasidisks and conformality II

Changyu Guo

Abstract

We introduce a weaker variant of the concept of three point property, which is equivalent to a non-linear local connectivity condition introduced in [12], sufficient to guarantee the extendability of a conformal map f from the unit disk onto a domain to the entire plane as a homeomorphism of locally exponentially integrable distortion. Sufficient conditions for extendability to a homeomorphism of locally p-integrable distortion are also given.

Generalized quasidisks and conformality II

Abstract

We introduce a weaker variant of the concept of three point property, which is equivalent to a non-linear local connectivity condition introduced in [12], sufficient to guarantee the extendability of a conformal map f from the unit disk onto a domain to the entire plane as a homeomorphism of locally exponentially integrable distortion. Sufficient conditions for extendability to a homeomorphism of locally p-integrable distortion are also given.

Paper Structure

This paper contains 9 sections, 12 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Definitions
  3. Auxiliary results
  4. Extension of a conformal welding
  5. Main proofs
  6. Concluding remarks
  7. Definition of generalized quasidisks
  8. Inward pointing and outward pointing cusps
  9. Open problems

Key Result

Theorem 1.1

If a Jordan domain $\Omega\subset\mathbb{R}^2$ has the three point property with the control function $\psi(t)=Ct\log^{\frac{1}{2}}\frac{1}{t}$ for some positive constant $C$, then $\Omega$ is a generalized quasidisk.

Figures (1)

  • Figure 1: The homeomorphism $\tilde{G}$

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2: Lemma 3.5, gk12
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 7 more