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On a minimal complex norm that extends the real Euclidean norm. (English) Zbl 0638.32005

The authors construct a complex norm \(N^*\) that extends the real euclidean norm and prove that it is the smallest complex norm in the following sense: If N is any complex norm in \({\mathbb{C}}^ n\) which coincides with the real euclidean norm \(| \cdot |\) in \({\mathbb{R}}^ n\) and N(z)\(\leq | z|\) for \(z\in {\mathbb{C}}^ n\), then \(N^*(z)\leq N(z)\) for \(z\in {\mathbb{C}}^ n.\)
Let \(B^*_ n\) be the unit ball with respect to \(N^*\). Then \(B_ n^*\) is a convex complete circular domain with only a continuous boundary for \(n>1\). The authors obtain some complex geometric properties of \(B^*_ n:\) \(B^*_ n\) is neither biholomorphically equivalent to the unit ball \(B^ n\) nor the polydisc \(\Delta^ n\). In fact, \(B^*_ n\) is not even homogeneous. In particular, if \(n=2\), \(B^*_ 2\) is biholomorphically equivalent to \(D=\{z\in {\mathbb{C}}^ 2:| z_ 1| +| z_ 2| <1\},\) a rigid domain studied earlier by N. Kritikos [Math. Ann. 99, 321-341 (1928), JFM 54.0373.02].
Reviewer: K.T.Hahn

MSC:

32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32F45 Invariant metrics and pseudodistances in several complex variables
32H99 Holomorphic mappings and correspondences

Citations:

JFM 54.0373.02
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References:

[1] Kritikos, N.: Über analytische Abbildungen einer Klasse von vierdimensionalen Gebieten. Math. Ann.99, 321-341 (1928). · JFM 54.0373.02
[2] Rosay, J. P.: Sur une characterization de la boule parmi des domaines de ? n par son groupe d’automorphismes. Ann. Inst. Fourier, Grenoble29, 91-97 (1979). · Zbl 0402.32001
[3] Rudin, W.: Function Theory in the Unit Ball of ? n . Berlin-Heidelberg-New York: Springer. 1980. · Zbl 0495.32001
[4] Sadullaev, A.: Extremal plurisubharmonic function for the unit ball (Russian). Ann. Pol. Math.46, 433-437 (1985). · Zbl 0606.31005
[5] Siciak, J.: Extremal plurisubharmonic functions in ? n . Ann. Pol. Math.39, 175-211 (1981). · Zbl 0477.32018
[6] Vesentini, E.: Variations on a theme of Carathéodory. Ann. Scuola Norm. Sup. Pisa7, 39-68 (1979). · Zbl 0413.46039
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