We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for C 1,1 -smooth convex domains in C 2 which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for "generic" product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, nonpseudoconvex, Gromov hyperbolic domains in C n are given.

We consider a proper holomorphic map π : D → G between domains in C n and show that it induces a unitary isomorphism between the Bergman space A 2 (G) and some subspace of A 2 (D). Using this isomorphism we construct orthogonal projection onto that subspace and we derive Bell's transformation formula for the Bergman kernel under proper holomorphic mappings. As a consequence of the formula we get that the tetrablock is not a Lu Qi-Keng domain.for g ∈ A 2 (D), where {π j } m j=1 are the local inverses to π.(1) 2010 Mathematics Subject Classification. Primary: 32A25, 32A70, Secondary: 32M15.

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