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3 Annales Polonici Mathematici

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Kobayashi-Royden vs. Hahn pseudometric in ℂ²

100%

Jarnicki W.

2000

tom 75

nr 3

289-294

For a domain D ⊂ ℂ the Kobayashi-Royden ϰ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for $D ⊂ ℂ^n$, n ≥ 3, we have $h_D ≡ ϰ_D$. Let D₁, D₂ ⊂ ℂ. The aim of this paper is to show that $h_{D₁ × D₂}$ iff at least one of D₁, D₂ is simply connected or biholomorphic to ℂ \ {0}. In particular, there are domains D ⊂ ℂ² for which $h_D ≢ ϰ_D$.

Concave domains with trivial biholomorphic invariants

63%

Jarnicki W. , Nikolov N.

2002

tom 79

nr 1

63-66

It is proved that if F is a convex closed set in ℂⁿ, n ≥2, containing at most one (n-1)-dimensional complex hyperplane, then the Kobayashi metric and the Lempert function of ℂⁿ∖ F identically vanish.

On extremal holomorphically contractible families

51%

Jarnicki M. , Jarnicki W. , Pflug P.

2003

tom 81

nr 2

183-199

We prove (Theorem 1.2) that the category of generalized holomorphically contractible families (Definition 1.1) has maximal and minimal objects. Moreover, we present basic properties of these extremal families.