On the Gromov non-hyperbolicity of certain domains in Cn

Gromov hyperbolicity of the Kobayashi metric on C-convex domains

“…Let us recall that the statement holds for convex domains (cf. [8,Proposition 4]). Hence, by the holomorphic contractibility of the Kobayashi distance, we get the upper estimate for k Ω (p s , p t ).…”

“…Hence, the case considered by Haggui and Chrih is the essential one. One of our goal is to refine the construction used in [8], and show that what is really needed in Theorem 1.2 is a C-convexity.…”

“…This simple observation allows us to apply the construction of quasitriangles considered in [8]. However, Theorem 1.3 does not follow directly from [8].…”

“…In [8] Haggui and Chrih also investigated some Hartogs type domains. In the present paper we easily obtain the following generalization of [8, Theorem 3.2]: endowed with the Kobayashi distance is not Gromov hyperbolic.…”

“…The key ingredient in the proof of Theorem 1.3 is that every weakly linearly convex domain (and so C-convex) contains a "sufficiently" big convex set. This simple observation allows us to apply the construction of quasitriangles considered in [8]. However, Theorem 1.3 does not follow directly from [8].…”