Hyperbolizing hyperspaces
The aim of this paper is to establish connections between a metric space \(X\) and the large-scale geometry (in the sense of Gromov) of the hyperspace \(\mathcal H(X)\) of its non-degenerate closed bounded subsets, and to study the mappings on \(X\) in terms of the induced mappings on \(\mathcal H(X)\). The metric space \(X\) can be identified with the boundary of \(\mathcal H(X)\) when the latter is equipped with the Hausdorff metric, but stronger relationships between \(X\) and \(\mathcal H(X)\) are obtained when the hyperspace \(\mathcal H(X)\) is hyperbolized and the space \(X\) is identified with its boundary at infinity: a priori weak conditions on \(\mathcal H(X)\) are strengthened at the boundary at infinity. The basic tool for studying such relationships is Gromov's theory of negatively curved spaces. Let us consider Euclidean space \(\mathbb R^n\). The one-point compactification \(\overline{\mathbb R^n}\) is equipped with the chordal metric, which is Möbius equivalent to the Euclidean metric when restricted to \({\mathbb R^n}\). The space \(\overline{\mathbb R^n}\) can be identified with the ideal boundary of the hyperbolic space \(\mathbb H^{n+1}\) and, as such, it inherits a family of visual metrics from \(\mathbb H^{n+1}\). In fact, the chordal metric is one such visual metric. The Möbius transformations of \(\overline{\mathbb R^n}\) can be extended to isometries of \(\mathbb H^{n+1}\) by the Poincaré extension. The quasisymmetric maps of \(\mathbb R^n\) can be extended to bi-Lipschitz maps of \(\mathbb H^{n+1}\) by the Tukia-Väisälä extension operator and to quasi-isometries of \(\mathbb H^{n+1}\) by the Tukia extension operator. In this paper, the author shows that connections analogous to those just described between \(\mathbb R^n\) and \(\mathbb H^{n+1}\) can be established between a more general metric space and its hyperspace. More precisely, the author replaces the space \(\mathbb R^n\) with an arbitrary complete perfect metric space \(X\). The one-point extension \({\widehat X} = X\cup\{\infty\}\) of \(X\) is endowed with a family of chordal metrics whose restrictions to \(X\) are (quasi-)Möbius equivalent to the original metric of \(X\). The hyperbolic space \(\mathbb H^{n+1}\) is replaced with the hyperspace \(\mathcal H(X)\) equipped with a metric \(d_{\mathcal H}\). It is shown that the metric \(d_{\mathcal H}\) induces the same topology on \({\mathcal H}(X)\) as the one induced by the Hausdorff metric, that the space \(\big({\mathcal H}(X),d_{\mathcal H}\big)\) is Gromov hyperbolic and its boundary at infinity can be identified with \(\widehat X\), and that the chordal metrics on \(\widehat X\) are visual metrics. In particular, the space \(\big({\mathcal H}(\mathbb R^n),d_{\mathcal H}\big)\) is roughly isometric to the hyperbolic space \(\mathbb H^{n+1}\). The author further shows that if \(f\) is a quasi-symmetric map between arbitrary metric spaces \(X\) and \(Y\), then the extension map \(\hat f\) is a quasi-isometry between \(\big({\mathcal H}(X),d_{\mathcal H}\big)\) and \(\big({\mathcal H}(Y),d_{\mathcal H}\big)\). Moreover, the quasi-isometry constant of \(\hat f\) depends only on the quasi-symmetry constant of \(f\).
quasi-isometric embeddings of hypersurfaces
scientific article; zbMATH DE number 5899032
scientific article