# Geometric characterizations of inner uniformity through Gromov hyperbolicity

## TL;DR

This paper establishes the equivalence of three geometric conditions characterizing inner uniformity of domains in Euclidean space using Gromov hyperbolicity and boundary properties, answering longstanding questions in geometric analysis.

## Contribution

It proves the equivalence of inner uniformity with Gromov hyperbolicity and boundary quasisymmetry, providing new insights into the geometric structure of domains.

## Key findings

- Inner uniformity is equivalent to Gromov hyperbolicity plus boundary quasisymmetry.
- Gromov hyperbolic domains with linearly locally connected inner metrics satisfy inner uniformity.
- Answers to three open questions from Bonk, Heinonen, and Koskela (2001).

## 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.