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Geodesics for convex complex ellipsoids. (English) Zbl 0812.32010
Denote by \({\mathcal E} (p)\) the complex ellipsoid \(\{(z_ 1, \dots, z_ n) \in \mathbb{C}^ n |\;\sum^ n_{j=1} | z_ j |^{2p_ j} < 1\}\). This paper describes all complex geodesics \(\varphi:E\to{\mathcal E}(p)\), where \(E:=\{\lambda\in\mathbb{C}|\;|\lambda|<1\}\). The holomorphic mapping \(\varphi : E \to {\mathcal E} (p)\) is a complex geodesic if it preserves the Kobayashi distance.
Remark. A simplification of the proof of the main theorem is in the preprint by the first two authors entitled “Geodesics for convex complex ellipsoids. II”.

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