An estimate for the quasiconformality coefficient of a domain via the curvature of its quasihyperbolic metric (Q1307185)

It is well known that, in the Poincaré model (that is, in the half-space \(x_n >0\)), the metric of the hyperbolic (or Lobachevskij) space is given by the formula \[ ds^2=\frac{dx_1^2 + dx_2^2 +\dots +dx_n^2}{x_n^{2}}. \] Similarly, the metric, defined by the formula \[ ds^2=\frac{dx_1^2 + dx_2^2 +\dots +dx_n^2}{\rho (x)^{2}} \] in a domain \(D\subset \mathbb R^n\), is called quasihyperbolic. Here \(\rho (x)\) stands for the Euclidean distance between \(x\in D\) and the boundary of \(D\). In the article under review, the author shows that the notion of one-dimensional sectional curvature can be introduced for the quasihyperbolic metric (as well as for any conformally flat metric). Therewith it is shown that \(D\) is convex if and only if the one-dimensional sectional curvature of its quasihyperbolic metric is less than or equal to \(-1/2\). Besides, the author constructs some special quasihyperbolic metrics in \(D\) which are invariant under Möbius changes of variables. This makes it possible to characterize domains \(D\), each of which can be taken to a convex domain by a suitable Möbius transformation, in terms of one-dimensional sectional curvature of the above-mentioned special quasihyperbolic metrics in \(D\). This result is related to results previously obtained by \textit{D. E. Blair} and \textit{J. B. Wilker} in [Kodai Math. J. 6, 186-192 (1983; Zbl 0524.53002)]. Furthermore, the author calculates the maximal value of the one-dimensional sectional curvature of the quasihyperbolic metric of a channel surface and applies the technique developed to the theory of mappings with bounded distortion.