Gromov hyperbolicity is a large-scale negative curvature notion for metric spaces, generalizing the properties of the hyperbolic plane and of metric trees. An important problem in several complex variables is to determine which domains of C^n are Gromov hyperbolic w.r.t. their Kobayashi distance. Balogh-Bonk proved that this is the case for strongly pseudoconvex domains, and Zimmer proved it for convex domains of D’Angelo finite type.

On the other side it is known (Gaussier-Seshadri) that an analytic disc in the boundary is an obstruction to Gromov hypebrolicity for a convex domain. In this talk I will discuss the case of the Worm domain, which is a non-convex domain with an analytic annulus in the boundary. I will show that it is not Gromov hyperbolic using a scaling method. This is a work in collaboration with G. Dall’Ara and M. Fiacchi.