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Characterizations of the unit ball \(B^n\) in complex Euclidean space. (English) Zbl 0547.32013
Two characterizations are given. It is shown that if \(M\) is a complex manifold such that at one point of \(M\) the Eisenman-Kobayashi and Eisenman- Carathéodory volume forms coincide and are nonzero, then \(M\) is biholomorphic tothe unit ball \(B^n\). This generalizes previous results of B. Wong, Rosay, and Dektyarev. It is also shown, using results of Lempert and Royden - P.M. Wong, that if \(\Omega\) is a strictly convex domain in \({\mathbb{C}}^ n\) such that either (a) there exists a point \(p\in\Omega\) and a holomorphic map \(\tau: B^n\to\Omega\), \(\tau(0)=p,\) such that \(d\tau(0)\) carries the unit ball onto the indicatrix of the Kobayashi metric; or (b) there exists a point \(p\in\Omega \) and a holomorphic map \(\sigma:\Omega\to B^n\), \(\sigma(p)=0\), such that \(d\sigma(p)\) carries the indicatrix of the Carathéodory metric onto the unit ball, then \(\Omega\) is biholomorphic to \(B^n\).

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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References:
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