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Geometric characterizations of inner uniformity through Gromov hyperbolicity

Manzi Huang, Antti Rasila, Xiantao Wang, Qingshan Zhou

TL;DR

This paper resolves the Characterization Question for inner uniformity in bounded domains of $\mathbb{R}^n$ by proving the three equivalent conditions: (i) inner uniformity, (ii) Gromov hyperbolicity with the inner boundary $\partial_\sigma G$ naturally quasisymmetric to the Gromov boundary $\partial^*G$, and (iii) Gromov hyperbolicity with linear local connectedness in the inner metric $\sigma$. The authors develop a framework combining inner geometry, Gromov hyperbolicity, conformal deformations, and boundary mappings, introducing Condition A and Condition B to bridge interior properties with boundary data. They establish a diameter-version of quasiconvexity under Condition B, prove inner uniformity from Condition B, and then complete the circle by showing LLC implies inner uniformity via Gehring–Hayman and ball-separation criteria. The results answer longstanding open questions (notably from Bonk–Heinonen–Koskela) and extend characterizations of uniformity to the inner metric, with implications for boundary regularity and geometric function theory in non-Euclidean settings.

Abstract

In this paper, we study the characterization of inner uniformity of bounded domains $G$ in $\IR^n$, and prove that the following three conditions are equivalent: $(1)$ $G$ is inner uniform; $(2)$ $G$ is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; $(3)$ $G$ is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions $(1)$ and $(2)$, and the implication from $(2)$ to $(3)$ affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.

Geometric characterizations of inner uniformity through Gromov hyperbolicity

TL;DR

This paper resolves the Characterization Question for inner uniformity in bounded domains of by proving the three equivalent conditions: (i) inner uniformity, (ii) Gromov hyperbolicity with the inner boundary naturally quasisymmetric to the Gromov boundary , and (iii) Gromov hyperbolicity with linear local connectedness in the inner metric . The authors develop a framework combining inner geometry, Gromov hyperbolicity, conformal deformations, and boundary mappings, introducing Condition A and Condition B to bridge interior properties with boundary data. They establish a diameter-version of quasiconvexity under Condition B, prove inner uniformity from Condition B, and then complete the circle by showing LLC implies inner uniformity via Gehring–Hayman and ball-separation criteria. The results answer longstanding open questions (notably from Bonk–Heinonen–Koskela) and extend characterizations of uniformity to the inner metric, with implications for boundary regularity and geometric function theory in non-Euclidean settings.

Abstract

In this paper, we study the characterization of inner uniformity of bounded domains in , and prove that the following three conditions are equivalent: is inner uniform; is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions and , and the implication from to affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.

Paper Structure

This paper contains 22 sections, 29 theorems, 172 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform domains, inner uniform domains and John domains
  4. Quasihyperbolic metrics
  5. Quasigeodesics and quasigeodesic rays
  6. Conformal deformations, Gromov hyperbolic domains and Gromov hyperbolic spaces
  7. Geometric characterization of Gromov hyperbolicity
  8. Visual metrics and natural mappings
  9. Useful classes of mappings
  10. Notational conventions
  11. Rough starlikeness
  12. Two conditions
  13. Proofs of the implications $(\mathfrak{a})$ $\Longrightarrow$ $(\mathfrak{b})$ $\Longrightarrow$ $(\mathfrak{c})$ in Theorem \ref{['thm-2.3']}
  14. Auxiliary lemmas
  15. Several constants
  16. ...and 7 more sections

Key Result

Theorem 1.1

Suppose that $G$ is a bounded domain in ${\mathbb R}^n$ with $n\geq 2$. Then the following conditions are equivalent:

Figures (2)

  • Figure 1: The related points and curves.
  • Figure 2: The related points and curves.

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Theorem 3.1
  • Lemma 3.1
  • Remark 4.1
  • ...and 22 more