Uniformizing Gromov hyperbolic spaces (Q2715761)

A geodesic metric space \(X\) is Gromov hyperbolic if for some \(\delta\) for all triples of geodesics \([x,y]\), \([y,z]\), \([z,x]\) in \(X\) every point in \([x,y]\) is within distance \(\delta\) from \([y,z]\cup [z,x]\). A locally compact, rectifiably connected noncomplete space \(\Omega\) is called \(A\)-uniform if \(\Omega\) is a uniform domain as in the classical euclidean case [\textit{O. Martio}, \textit{J. Sarvas}: Ann. Acad. Sci. Fenn., Ser. A I 4, 383-401 (1978; Zbl 0427.30021)] except that the boundary \(\partial\Omega\) of \(\Omega\) is defined as \(\partial\Omega=\overline\Omega \setminus\Omega\) where \(\overline \Omega\) denotes the completion of \(\Omega\). In this study of conformal and quasiconformal analysis on metric spaces the two aspects of the unit disk as a typical plane uniform domain and as a complete Riemannian 2-manifold of constant negative curvature are replaced by a general \(A\)-uniform space \(\Omega\) and a Gromov hyperbolic space \(X\), respectively. After Introduction and basic definitions the authors prove the uniformization theorem that there is a one-to-one correspondence between the quasiisometry classes of proper geodesic roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of \(A\)-uniform spaces. They then continue to study in the metric set-up the Gehring-Hayman [\textit{F. W. Gehring}, \textit{W. Hayman}: J. Math. Pures Appl., IX Sér. 41, 353-361 (1962; Zbl 0105.28002)] theorem that the hyperbolic geodesic almost minimizes the Euclidean length among all the curves in the domain with the same end points. The rest of the chapters are devoted to the Loewner property, to Gromov hyperbolic spherical domains, to the Martin boundary of Gromov hyperbolic spaces and to quasiconformal maps between these spaces. The exposition greatly clarifies the assumptions and conditions on metric spaces in order to have useful conformal or quasiconformal analysis available on these spaces.