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<p>In this paper, we study the characterization of inner uniformity of bounded domains <i>G</i> in ℝ<sup><i>n</i></sup>, and prove that the following three conditions are equivalent: (1) <i>G</i> is inner uniform; (2) <i>G</i> is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; (3) <i>G</i> 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.</p>

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Geometric characterizations of inner uniformity through Gromov hyperbolicity

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  • 发表日期：

    2026年4月19日

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  • Manzi Huang,
  • Antti Rasila,
  • Xiantao Wang,
  • Qingshan Zhou

摘要

In this paper, we study the characterization of inner uniformity of bounded domains G in ℝn, 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.