Hyperbolic metrics on open subsets of Ptolemaic spaces with sharp parameter bounds

Abstract

It is shown that a construction of Z. Zhang and Y. Xiao on open subsets of Ptolemaic spaces yields, when the subset has boundary containing at least two points, metrics that are Gromov hyperbolic with parameter log ⁡ 2 \log 2 and strongly hyperbolic with parameter 1 1 with no further conditions on the open set. A class of examples is constructed on Hadamard manifolds showing that these estimates of the parameters are sharp.