Hyperbolic Type Metrics

The natural setup for our work here is a metric space (G, m G) where G is a subdomain of ℝ n , n ≥ 2 $$\mathbb {R}^n\,, n\ge 2$$ . For our studies, the distance m G(x, y), x, y ∈ G is required to take into account both how close the points x, y are to each other and the position of the points relative to the boundary ∂G. Metrics of this type are called hyperbolic type metrics and they are substitutes for the hyperbolic metric in dimensions n ≥ 3. The quasihyperbolic metric and the distance ratio metric are both examples of hyperbolic type metrics. A key problem is to study a quasiconformal mapping between metric spaces f : ( G , m G ) → ( f ( G ) , m f ( G ) ) $$\displaystyle f: (G, m_G) \to (f(G), m_{f(G)}) $$ and to estimate its modulus of continuity. We expect Holder continuity, but a concrete form of these results may differ from metric to metric. Another question is the comparison of the metrics to each other.