Uniform domains and hyperbolic distance
The concept of a uniform domain in \(\mathbb R^n\) was introduced in [the reviewer and \textit{J. Sarvas}, Ann. Acad. Sci. Fenn., Ser. A I, Math. 4, 383--401 (1979; Zbl 0406.30013)] and since then these domains are used if slightly less regular domains than smooth domains are needed. The uniformity concept has also been extended to more general metric spaces. Roughly speaking in a uniform domain \(\Omega\) every pair of points \(z_1, \, z_2\) can be joined by a rectifiable path \(\gamma\) whose length is comparable to dist\((z_1, z_2)\) and moves away from \(\partial \Omega\) comparable to \(\min(\mathrm{dist}(z_1, \gamma(s)), \mathrm{dist}(z_2, \gamma(s)))\). The author characterizes the uniformality of hyperbolic plane domains \(\Omega\) (i.e., the complement \(\mathbb R^2 \setminus \Omega\) has at least two points) in terms of the Poincaré hyperbolic length distance \(h\) and the quasihypebolic distance \(k_{\Omega}\) defined as \(k_{\Omega}(z_1,z_2) = \inf \int_{\gamma} \delta(\gamma(s))^{-1}\, ds\) where the infimum is taken over all paths \(\gamma\) which connect \(z_1\) to \(z_2\) in \(\Omega\) and \(\delta(z) = \mathrm{dist}(z, \partial \Omega)\). For the characterization the so called relative distance \(j(z_1, z_2) = \log(1 + |z_1 - z_2|/\min(\delta(z_1),\delta( z_2)))\) in \(\Omega\) is used and it is shown that uniformity, \(k_{\Omega} \leq C_1\, j \) and \(h \leq C_2 \, j\) are equivalent and the constants depend only on each other. The relations between uniformity and quasihyperbolic distance have been previously known in rather general metric spaces but the proof from \(h \leq C_2j\) to uniformity is difficult. If instead of the usual distance the inner distance \(\ell(z_1,z_2) = \inf \{\mathrm{lenght}(\gamma): \, \gamma \text{ joins }z_1\text{ to }z_2 \text{ in }\Omega\}\) replaces \(|z_1 - z_2|\) in the definition of \(j\), then the equivalence still holds. Various concepts related to uniformity and their use in general metric spaces are carefully analysed in the paper.
scientific article; zbMATH DE number 7382538
scientific article