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The Lempert theorem and the tetrablock. (English) Zbl 1279.32011

The paper under review is a continuation of the study of the geometry of the tetrablock, a domain in \(\mathbb C^3\) defined by \[ \mathbb E:=\big\{(z_1,z_2,z_3)\in\mathbb C^3:|z_2-\bar z_1z_3|+|z_1z_2-z_3|+|z_1|^2<1\big\}. \]
A. A. Abouhajar, M. C. White and N. J. Young proved in [J. Geom. Anal. 17, No. 4, 717–749 (2007; Zbl 1149.30020)] that the equality between the Carathéodory distance and the Lempert function of \(\mathbb E\) with one of the arguments fixed at the origin holds on \(\mathbb E\), and asked whether such an equality holds on \(\mathbb E\times\mathbb E\).
The main result of the paper under review is to show that the equality between the Carathéodory distance and the Lempert function of \(\mathbb E\) holds on \(\mathbb E\times\mathbb E\).
It is also shown that \(\mathbb E\) cannot be exhausted by domains biholomorphic to convex ones.
In the proof of the main result, the authors make use of the following result, which is interesting in its own right:
Let \(D\) be an \((m_1,\dots,m_n)\)-balanced pseudoconvex domain. Assume that \(\psi\) is a complex geodesic in \(D\) and \[ \psi(\lambda)=(\lambda^{m_1}\varphi_1(\lambda),\dots,\lambda^{m_n}\varphi_n(\lambda)),\quad\lambda\in\mathbb D, \] for some \(\varphi_j\) holomorphic on \(\mathbb D:=\{\lambda\in\mathbb C:|\lambda|<1\}\), \(j=1,\dots,n\). Then \(\varphi=(\varphi_1,\dots,\varphi_n)\in\mathcal O(\mathbb D,\partial D)\) or \(\varphi\) is a complex geodesic in \(D\).

MSC:

32F45 Invariant metrics and pseudodistances in several complex variables

Citations:

Zbl 1149.30020
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References:

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