Gromov (non)hyperbolicity of certain domains in ${\mathbb{C}}^2$

Abstract

We prove the non-hyperbolicity of the Kobayashi distance for ${\mathcal{C}}^{1,1}$-smooth convex domains in ${\mathbb{C}}^2$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. Moreover, examples of smooth, non pseudoconvex, Gromov hyperbolic domains are given; we prove that the symmetrized polydisc and the tetrablock are not Gromov hyperbolic and write down some results about Gromov hyperbolicity of product spaces.

17 pages ; typos corrected, some proofs rewritten for clarity following the referee's comments. To appear in Forum Mathematicum

References in corpus (1)

• The Kobayashi balls of C-convex domains

Cited by in corpus (2)

• On the Gromov hyperbolicity of the minimal metric
• Gromov hyperbolicity of pseudoconvex finite type domains in ${\mathbb{C}}^2$