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)


32F45 Invariant metrics and pseudodistances in several complex variables


Zbl 0847.32027
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