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arXiv · 1706.05494

Geometric characterizations of inner uniformity through Gromov hyperbolicity

Manzi Huang·Antti Rasila·Xiantao Wang·Qingshan Zhou
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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.

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Manzi Huang, Antti Rasila, Xiantao Wang, Qingshan Zhou. 2017-06-17. Geometric characterizations of inner uniformity through Gromov hyperbolicity. https://arxiv.org/abs/1706.05494

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