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Kobayashi-Royden vs. Hahn pseudometric in $${\mathbb C}^2$$. (English) Zbl 0982.32012
For a domain in $$D \subset \mathbb{C}^n$$, the Kobayashi-Royden pseudometric $$k_D$$ and the Hahn pseudometric $$h_D$$ are defined, by the formulas $k_D(z; X) =\inf\{|\alpha|:f(0)=z, \alpha f'(0) = X\},$
$h_D(z; X) =\inf\{|\alpha|:f(0)=z, \alpha f'(0) = X, f \text{ is injective}\},$ for $$z \in D, X \in \mathbb{C}^n$$, here the inf is taken for holomorphic maps $$f: \bigtriangleup \rightarrow D$$, where $$\bigtriangleup$$ is the unit disc. Obviously $$k_D \leq h_D$$.
It is known that for a domain $$D \subset \mathbb{C}$$, $$k_D = h_D$$ if and only if $$D$$ is simply connected. Overholt showed that for any domain $$D \subset \mathbb{C}^n$$, with $$n \geq 3$$, $$k_D = h_D$$ [M. Overholt, Ann. Pol. Math. 62, No. 1, 79-82 (1995; Zbl 0847.32027)]. This paper proves the following result: Let $$D_1, D_2 \subset {\mathbf C}$$ be domains. Then: (1) If at least one of $$D_1, D_2$$ is simply connected, then $$k_{D_1\times D_2} \equiv h_{D_1\times D_2}$$. (2) If at least one of $$D_1, D_2$$ is biholomorphic to $$\mathbb{C}_*$$, then $$k_{D_1\times D_2} \equiv h_{D_1\times D_2}$$. (3) Otherwise $$k_{D_1\times D_2} \not\equiv h_{D_1\times D_2}$$.
Reviewer: Min Ru (Houston)

MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables
Zbl 0847.32027
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