EUDML

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Kobayashi-Royden vs. Hahn pseudometric in ℂ²

Witold Jarnicki — 2000

Annales Polonici Mathematici

For a domain D ⊂ ℂ the Kobayashi-Royden ϰ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for $D \subset ℂ^{n}$ , n ≥ 3, we have $h_{D} \equiv ϰ_{D}$ . Let D₁, D₂ ⊂ ℂ. The aim of this paper is to show that $h_{D ₁ \times D ₂}$ 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

Witold Jarnicki; Nikolai Nikolov — 2002

Annales Polonici Mathematici

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

Marek Jarnicki; Witold Jarnicki; Peter Pflug — 2003

Annales Polonici Mathematici

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.