On the Representative Domain

  • Qi-keng Lu

Abstract

Professor Stefan Bergman [1] introduced the idea of Repräsentantenbereich of a bounded domain Open image in new window in ℂn. However the strict definition of a representative domain is not very clear as noted in [2]. It seems that he called the image f( Open image in new window ) of the mapping f:

$$ {\text{z}} \to \frac{{\partial \log \frac{{{\text{k}}({\text{z}},\bar t)}} {{{\text{k}}({\text{t}},\bar t)}}}} {{\partial \bar t^\beta }}{\text{T}}^{\bar \beta \alpha } ({\text{t}},\bar t) $$
(1)
the representative domain of Open image in new window . Here we use the summation convention and denote by \( {\text{k}}({\text{z}},\bar t) \) the Bergman kernel of Open image in new window and by \( T^{\overline \beta \alpha } (t,\overline t ) \) the elements of the inverse matrix of the Bergman metric tensor
$$ T_{\alpha \overline \beta } (t,\overline t ) = \frac{{\partial ^2 \log \,K(t,\overline t )}}{{\partial t^\alpha \partial \overline t ^\beta }}.$$

Keywords

Manifold 

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References

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    Bergman, S., Über die Existenz von Repräsentantenbereichen in Theorie der Abbildung durch Paare von Funktionen zweier komplexen Veränderlichen. Math. Ann. 102 (1929), 430–446.CrossRefGoogle Scholar
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Copyright information

© Birkhäuser Boston, Inc. 1984

Authors and Affiliations

  • Qi-keng Lu
    • 1
  1. 1.Institute of MathematicsAcademia SinicaChina

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