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We introduce the notion of locally visible and locally Gromov hyperbolicdomains in $\mathbb C^d$. We prove that a bounded domain in $\mathbb C^d$ islocally visible and locally Gromov hyperbolic if and only if it is (globally)visible and Gromov hyperbolic with respect to the Kobayashi distance. Thisallows to detect, from local information near the boundary, those domains whichare Gromov hyperbolic and for which biholomorphisms extend continuously up tothe boundary.

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