Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

We introduce the notion of locally visible and locally Gromov hyperbolic domains in C^n. We prove that a bounded domain in C^n is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to detect, from local information near the boundary, those domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary.