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Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains. (English) Zbl 1195.30061

A Denjoy domain is a domain in the complex plane whose boundary is contained in the real axis. Gromov hyperbolicity can be formulated by requiring all geodesic triangles to be thin. The main result of the paper under review is that a Denjoy domain is Gromov hyperbolic with respect to the hyperbolic metric if and only if it is Gromov hyperbolic with respect to the quasihyperbolioc metric. Whether this is true for a wider class of domains remains an open question.
Related work by the authors is carried out in [Complex Var. Elliptic Equ. 55, No. 1–3, 127–135 (2010; Zbl 1190.30030)], but the main difference here is the study of quasigeodesics instead of geodesics. The authors show several different classes of curves are quasigeodesics for the hyperbolic and quasihyperbolic metrics, and exhibit the relevant constants. Finally, as an application, the authors consider some concrete examples of domains which are, or fail to be, Gromov hyperbolic.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
30C99 Geometric function theory

Citations:

Zbl 1190.30030
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