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Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality

Pawel Goldstein, Zofia Grochulska, Chang-Yu Guo, Pekka Koskela, Debanjan Nandi

Abstract

In this paper, we extend the characterization of John disks obtained by Näkki and Väisälä [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by Väisälä in his study of metric duality theory [Math. Scan. 1997]. Somewhat surprisingly, we constructed a uniform domain in $\R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three.

Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality

Abstract

In this paper, we extend the characterization of John disks obtained by Näkki and Väisälä [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by Väisälä in his study of metric duality theory [Math. Scan. 1997]. Somewhat surprisingly, we constructed a uniform domain in , which is topologically simple, such that the complementary domain fails to be homotopically -bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three.

Paper Structure

This paper contains 4 sections, 19 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and Definitions
  3. Proof of Theorem \ref{['thm main: characterization']}
  4. Proof of Theorem \ref{['thm main: example']}

Key Result

Proposition 1.1

nv91 Let $\Omega\subset \dot{\mathbb{R}}^2$ be a simply connected domain such that $\overline{\Omega}$ does not contain the point at infinity. Then the following conditions are quantitatively equivalent:

Figures (3)

  • Figure 1: Example \ref{['example:easy one']}
  • Figure 2: A pair of 'horns'
  • Figure 3: Alexander's horned ball (2nd stage of the construction)

Theorems & Definitions (31)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1: Metric Duality Theorem
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof : Proof of Theorem \ref{['thm main: characterization']}
  • Example 3.1
  • proof : Construction of Example \ref{['example:easy one']}
  • ...and 21 more