Hyperbolic imbeddedness and tautness modulo a closed subset.

In the present paper, we study some properties of the Kobayashi metric on unbounded convex domains and on Hartogs domains defined by Ω φ ( C m ) = { ( z , w ) ∈ C m × C : | w | < e − φ ( z ) } , m ∈ N ∗ , where φ is a smooth strictly plurisubharmonic function such that φ ( z ) → + ∞ as | z | → ∞ . [ABSTRACT FROM AUTHOR]